Equations Book
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@ ò î ö b — y ⁄ a @t ì z j Û a ë @k í Š ‡ n Ü Û @ï i ‹ È Û a @‡ è È ¾ a @
! " # $ % &
. ! ' ( ' ) ' (
+ ( " + , - . , / 0 . 1
2 3
4 , / 5 ' 6 7 " 1 8 9 1 : 2 5 , ) , 6 7 ' , ( ; ' 0 . '
+ ' < ( 8 < (
= , > 1 = ( , 7 ( ? -
8 9 1 # 1 . > @ 6 &
% 5 A #
+ 7 .( A , - # ) > 1 8 , + 6+
! < D ' . D - < (
) - (
. ( ,
.
.
1
2
PROBABILITIES
"
. " (E v e n t )
! " # ! $ % & ' % ( ) !
, & % % + ! (* ) % '
" - & ' $ . / 0 1 % $ 2 % $ $ . /
4 % % 5 , ! .+ " ' 3 & $ 1/4 . $ - (52) .
4# $ "& " $ -
$ $ - % , 9 % % . $ & ' : % % ; '
" 1/4 & ' & $ - < # = % > - % % % ? ' " 4#
.13 .
$ & '
"& "
@ . 1/ 4 =
5 , 4
13/52 . ' , 9
B % , ( . 2 > 2 0 % 2
, 9 1 5 # & ' 5 , $ / " % ! 3 3 " = - & ' 2 % " % ! % & 4# 4
& ' $ # $ C % ' D , . " % ! % ' & ' $ = E
& ' '
0 < 9 2 - " C & ' $ = E
+ # < 9 ? # ' E 4! & ' = ' 4 < # . $ #
& % (F E R M AT ) " ' (PASCAL ) ! ) " C
3
$
" & ' + $ % " % ! % . . (BE R N O U L L I ) . # & '
( % # # " / % 0 F
:
! % & !
: , . & ' E & F
: 1
! / * % (* ) % . & ! $ + ; % % % F
'
5 4 3 2 1 $ : @ / & . $ % ! H I %
K 5 3 1 J J ' - / % % F
1 % 6
< K K K K. & K $ K # ; K ' , K ! . . $ K
! " < % . . J ' - / (E x p e r i m e n t ) < K / $ % ! " > $ (E v e n t ) .(S a m p l e S p a c e )
. & % L 1 ) ! $ ! - 3 / * + M
=
(P o s s i b l e C a s e s ) # 2
$ % / ! & $ 1 # E H I % " & .
% ! $ # $ - & % (* ) ' $ % $
5 4 3 2 1 % ! % . & % $ !
6 $ # $ - & $ & ' 2 $ % ! "
' 6 . % . & $ & '
(F a v o r a b l e C a s e s ) # 3 4
. J , 3 < 9 J N & " H I % & .
& ' J ' - < # = . 3 ! , ; ' % . > C & .
3 , .
& " ; ' % .
& $
" < $ ) * ) " D , . 5 3 1 < # =
.$
(E q u a l l y L i k e l y C a s e s ) $ " # 4
& ' $ % $ % = $ % " ! %
! , 9
% ! % 0 ! & ' . % C 0 1 % $ ' ) ! ! ! J $ # ) " ! " ! D , . ; ' ( # E
.4 & ' 4 = % 0 1 % %
(M u t u a l l y E x c l u s i v e E v e n t s ) % 5
. ) , 9 ' % % B A )
" - & ' < # = ! % & % (* ) '
.
(I n d e p e n d e n t E v e n t s ) $ & 6
. 9 B - ! , 9 # ) B A )
$ # $ - & % (* ) ' . E O B - & ' ) N : -
.< @ $ % ) ? $ % ) $ $ % ; ' (E x h a u s t i v e E v e n t s ) $ ' 7
! , 9 $ & ' $ # 2 3 ... C B A 3 <
.$ + 9 % . 9 3 5
! , 9 : $ ' $ 4 E % (* ) '
1 # : % @ $ # 2 3 " D , . E % E < # = ; ' 5 , ! ." 1 = D , . $ 1 =
: !
% & % 6 5 4 3 2 1
.. 9 3 : % @ $ # 2 3
" ( ) ( & * + & " < # $ 3 $ $ < # 1 =
$ 3
: D
: ! .B - ! @ 3
. J / % P
. & P
(T h e o r e t i c a l A p p r o a c h )
$ ) 3 J / % & ! * ! (C
F
' < # 3 $ % ! "
< 9 $ "
$ % . .3 & ' + & ' ! 4 = % " ! : 1, "
3 8
! " # $
11 = $ % ! "
3 = $ "
6
: ,
3/11 = + C ! 4 ∴
$ ' $ ) 3 4 , 9
. $ "
3 & ' + & ' ! 4 = % & $ % ! " :2 , "
. " K! 2 Q K9 9 0 # + C (4) 5 % . K! , 9
& ' $ ! 2 ) % % ) E 2 (D C B A)
: $ " N
A C E . .
D A C E . .4 D A C E . .R
A C E . .
J $ % ! " $ . S & % L 1 : , S = {AB, AC, AD, BC, BD, CD}
" 3 . A C E $ "
P ." 6 $ % ! " $ "
= $ % ! "
P(A ) =
3 1 = 6 2
D A E $ " 5 5 % . P(A or D) = P(AUD) =
7
5 6
P4
D , A E $ $ 5 % . PR P (A , D) = P (A ∩ D) =
1 6
A E $ " 3 5 % . P
∴ P (A ) = A E & .
3 1 = 6 2
P( A ) 3
* J / % & ' " = 4 F
% 9 4# 0
5 % .
# + ( (
$ % : # " , E SP %
! (* ) '
. K 4 +
& ' 0
* &
*
4% % # R 4 +
& ' 0
$ % B - 4
*
"
$ % B - 4
$ "
) & & . K 4 +
% B
$ % ! "
) & % B 4%
& % D , .
4 ( 4= : % Q % % % ! .* J / % 4= 8
:%
1
# , " - . + 2
, . ! $ % ! " ) : 2 J / % '
$ : $ ' S
E 2 E % (* ) ' I
% % ! * ' # R 4 & . (: )
4 ! 1/4 J T* ) (* : % !
U C : % @ J / % E , .
.$ # ) $ % ! " D , .
1/2 J ( ! , ! ? > % 5 , !
" ( ! , : % ! & . $ $ " < #
+ % > & ' . 2 : % @ 5 , (< ) % ! , ) $ % !
.3 % M < # ! ,
.
D , . ) < # 4# V * E W < 9 $ " ? 2 % % . .& . , . " =
(E m p i r i c a l A p p r o a c h )
! N " * J ! 2 < # " ! $ # % , 9 & "
n1 < # < 9 ! : / : /
% & "
% & "
n2 : /
%
; ' n3 V =
n1 = ! : # "
$ % N 9
n2 = : # "
$ % N n3 = V = : # "
$ % N 1 $ % ! " ) K # E 4% D , . 3
$ - % 4% D , . " - # ! ! N " % ! # !
.& < # % 10 % 20 % 70 (* ) ! - $ ) ! 3
: < # = .
L im
" ! $ # & % ! @
N
: < # = .
L im
" ! $ # & % @
N
: < # = .
L im
" ! $ # & % V = @
N
: /
n1 = 0.70 N → ∞
n2 = 0.20 N → ∞ n3 = 0.10 N → ∞
"
! n $ # $ - % , 9 5 , ! r 1/2 # E - $ % D , . . = =
$ % ; ' r . = n r $ % " - # ! n " # ! : % " ) ! V = n " % ! , 9 n
1/2 ( $ - $ % U = ( ! n ! % : % 3 1/2 L im N 10
n1 = 1/ 2 N → ∞
J
. % $ - & % =
< # = .
:
N
' r ! " # $ % & ( ) ! * + $ m
r
Li
. K 3 < % $ ' % 3 .... A3 , A2 , A1 3 " % ! , 9
J N K
D , . 3 ; ' & Q E @ 3 J 3 $ < 9
; ' ( ! ( 3
$ # ! ! ( * +
! , * + * ( ) (1) ! 2 & ' !
' % ) A B ! , ; '
(1)
s A
B
P (A B) = P(A) + P(B) ; '
) B - (C
P (AU B) = P (A) + P (B)
11
YJ '
< # = . % . & $ & ' : 3
3 5 3 1 < # = D % J '
< # = : ; ' $ ' % $ ) * ) 3 D , .
P (1 3 5 ) = P (1 ) + P (3 ) + P (5 )
1 1 1 + + 6 6 6
= =
1 2 :4
Y 2 < # = . % . & %
:
(2 2) (1 1) < # = D % 2 < # = 1 % ! $ ' % 3 & . (6 6) ... (3 3) 36 P ( 2 ) = P (1 1)+ P (2 2)+ ... + P (6 6)
1 1 1 + + ... + 36 36 36 6 = 36 =
12
% B, A P (A) + P(B) = 1
B, A P (A) = P (B) = 1/ 2
A A A # $ A ! " #
% &
% ' %
A) 3 = ! B A ) & ' % 2 %
B A ) B - 1 % < # B B - 1 % < # A B - (B
& (2) ! 2 U C ! " - & ' (2)
s
A and B
B
A
(' C A 3 # $ " B ) P (A) + P (B) O
K K! $ K/ * 4K ! B 3 # $ " B 9
$ K " C B 3 # $ 5 # A 3 # $ " > K % K% % ; ' P (B) P (A) > $ & ' : % ; ' , ( B A B - 13
= % P (A B ) [ , P (A B) : . , . P (A B) < #
P (A B )= P (A) + P (B) – P (A B)
P (AU B) = P (A) + P(B) – P (A ∩ B)
: 5
. " ! 2 Q 9 9 0 # + C ) < 9 B
Q 9 9 0 # + C 4 D C A C E Y " N & ' # ) (D C B A . ) " ! 2 :
$ % ! " B . & % L 1
{AB,
S =
AC, AD, BC, BD, CD}
(AB, AC, AD) & . A E $ " P
(AD, BD, CD) & . D E $ " P (AD) & . ( D A E $ " P
.( # % # = & $ % < # = % " > - 4 P (AU D) = P (A D) = P (A) + P (D) – P (A ∩ D) = 3 3 1 + − 6 6 6
14
:6
! "
< # = ) B
< # = ) A F
1 %
:
"J %
" ! $ % $ $ -
40 52 13 52
P (A) = P (B) =
P (A ∩ B) = P (AU B) =
10 52
" > % -
P (A) + P (B) – P (A ∩ B) 40 13 10 + − 52 52 52 43 52
= =
( ) * + , - *
+ . , $ / . , $ $ 0 1 2 .
; ' B A # ) %
! , 9
P (A ∩ B) = P (A) P (B) 15
: 7
Y % R & % (3 3) : < # = .
:
1 . % @ & Q 3 : < # = 6
1 . % & % ) & Q 3 : < # = 5 , ! 6 P (A ∩ B) =
P (A) P (B) = 1 1 1 × = 6 6 36
Conditional Probability
$ M "
, E " ! 2 +
- B E W () ! , 9 A 3 3 $ ' $
& I * 9 2 % - , E
( # ! (* ) ' 3 (= E 2 . $ ' ! : % ; ' 1 # & ' $ #
1 # & ' * M . 2 = - ! , 9 (A 3 ) $ # J 2 .(B 3 )
< B 3 B - 2 A 3 B - ; ' , ! .
A 3 P (A/B) =
16
< # 4! & 2
.B 3 B -
:& ) & 2 1 U C % : 8
+ C " ! 7 + " ! 3 < # J 0 !
4 < 9 % , ; ' . 9 D < # ! ! : % % % % F
1 %
$ ! " ; ' B + ! 4 < 9 A + C !
:& # ) # = % &
AA
C !
P
+ C $ % ) + < @
P
+ C $ % ) + C < @
AB BA
P
!
BB
P
C 9 ! 4 $ % ) ) < @ ! 4 H I % "
P(A/A) = 6/9
3 4 5 P(A) = 7/10
P(B) = 3/10
A
:& 5
A P(B/A) = 3/9 P(A/B) = 7/9
B
A
B P(B/B) = 2/9 17
B
=
:& # ! : ! 4! ; ' , ! .
+ C $ % ) + C < @ ! !
! 2 + C $ % ) ! & ' C + C < @ !
.+ C < @
P (A ∩ A) =
P (A , A) = P (A) P (A/ A) 7 6 42 × = = 10 9 90
P (A ∩ B) = P (A) P (B/ A) =
7 3 21 . = 10 9 90
P (B ∩ A) = P (B) P (A/ B) =
3 7 21 . = 10 9 90
P (B ∩ B) = P (B) P (B/ B) =
3 2 6 . = 10 9 90
K)
:& 2 * & % ) , . % O
. /
-
1 $ 2 3 ( ) P (B) B , A 0 #
. B ! * + . / A ! . /
P (A/B) = P
(A ∩ B) P(B)
:
( ) B ! * + . / A ! . / ( B ! B , A
18
4 ) + 3
P(A B) 4! H % % % % ! D * & 2 P (A ∩ B) = P (A) P (B/ A) = P (B) P (A/ B) ! 3 3 $ & 1 ' ) ) ! @ $ V = D , . % % % ! P (A ∩ B ∩ C) = P (A) P (B/ A) P (C/ AB) : 9
3 : % " + " ! 3 + C " ! 7 < # J 0 !
Y+ C # ! ! . 9 " !
< @ 4 & ' + C ! < # = .
& % ) 4 & ' + C ! < # = .
3 ) 4 & ' + C ! < # = . P (A1 ∩ A2 ∩ A3) =
:
A1
!
A2 A3
% # 2
P (A1) P (A2/ A1) P (A3/ A1A2)
=
=
7 6 5 10 9 8 210 720
19
: 1 0
. 9 4# $ - "
Y 2 !
< @ $ - & ' 10
< # = . $ % ) $ - & ' 10
< # = .
:
A
!
B
P (AB) = P (A) P(B/A) =
4 3 . 52 51
! 4 : " # 5 / 3 &
' . 7 # 6 # 6
J " 4 " # D , . ) = $ %
:& ) & ' $ C $ % 2 @ $ 3 : 1 1
:& # ! R $ ) * ) %
. % 2 %
' $ 2 6 I R
.$ % 3 %
' $ 2 8 II R
.& % 1 %
' $ 2 4 III R
20
$ I 2 $ ( 2 : % 4 $ I 2 $ E
.( 2 ! (P) . ' .(C :
$ $ I 2 $ # ) , . & ' . R $ ) * ) E %
. (B) ( (A) ( % ! 2 E %
( i)
(ii)
B ' ! & $ # D , . $ $ % 2 )
: 2 #
1 /3 1 /3
2 /6 4 /6
I
3 /8
II
1 /3
5 /8 1 /4
III 4 C
3 /4
A B
A B
A B
= ( $ " J 3 !
( 2 E ) II R E (* ) ' . , . & ' B ' ! " 1 3 . . ( % 3 8
B ! & 2 < 9 J N $ ' % " $ ) * ) : % 3
= J ) 4 # . $ ) * ) " D , . "
:(& 2 < #
21
P
1 4 1 5 1 3 . + . + . 3 6 3 8 3 4 =
1 4 5 3 + + 3 6 8 4
= =
1 49 49 = 3 24 72
(Baye s ' T h e ore m )
D 1 .$ $ # " 1 # E - % =
%
% C '
. % 4%
- % = ( I 2 " : . @ % = " - $ 1 D , . !
& $ / % - E % B % , . $ # I < # $ ] < # + % F
1 $ =
.& 2 * ( . !
" 4 < 9 $ / %
" # @ Q ! 4V & ' .$ $ % " # $ = E 2 : E * E 9 $ % $ 3 $ ' C 9
E < # & " .$ ? 3 , . Prio r
& C * E
& # < $ H I % < # = - $ = E 2
< # $ " " *
< # " (* ) ' Pro b a b il it y
% .& # * $ # ) # R% M $ ! -
J < $ ! $ % " # + C
< # 4
. Po s t e rio r Pro b a b il it y
22
% % . , & ' ! - > I 2 @ $ % (* ) ' :& . F
> I 2 @ !
1
^
A2
> I 2 @ !
^
^^
^^
^^
^^ ^^
^
^^
^^
^^ ^^
^^
^
^^
^ ^
10
^^
^^
A3
^^
^^ ^^
^^
'
A1
> I 2 @ !
2 3
1 D , . , $ = E F
^^
^^
^^ ^^
A10
.$ ' % $ # 2 $ ) " D , .
4% % . / D , . B $ % " % "
, 9
E ! - > .
< , . ; ' P(A4/B) F
% % % 9
' % # 2 ) ) ! @
.'
' % # 2 ) < # $ / %
S # : 2 $ $ # /
$ P (D)
≠ O
8
9 # -
0 B A
! : 2 ; 2 # $ ! ( D
P (A/ D) =
P ( A D) P(A D) + P(B D)
P (B/ D) =
P ( B D) P(A D) + P(B D)
S
D B A 23
A
A B % # C 1 $ %
& ' % E 0 . 40
: 12
! , 9
, 0. 30 # ) + % ! , 9 B B % # C 1 % -
$ I 2 $ % E , ; ' .B # C 1 , 0 . 40 A # C 1
YA B % # C 1 ! . ' " % ! % E :
:& . " F
A B % C 1 E
B B % C 1 E
C 1 . A B % 2 (*
C 1 . B B % 2 (*
E ! E !
C 1 . A B % 2 D E !
C 1 . B B % 2 D E !
1 %
P (A) P (B) P (M/A) P (M/B) P (F/A) P (F/B)
:& ! 2 C $ % ! " " M
P(M /A) = 7 /10
P(A) = 4/10
P(B ) = 6 /10
A
P (A F ) = P (A ) P (F /A ) = F
P(F /A) = 3 /10
P (A M ) = P (A ) P (M /A ) =
P(M /B ) = 6 /10
M
P(F /B ) = 4/10
F
4 3 . 10 10
=
28 100 12 100
=
P (B M ) = P (B ) P (M /B ) = 6 /1 0 . 6 /1 0 = 3 6 /1 0 0
B
24
4 7 . 10 10
P (B F ) = P (B ) P (F /B ) =
6 4 . 10 10 =
24 100
- E
P (A/F) =
P (AF) 12/100 = P(AF) + P (BF) 12/100 + 24/100 =
6
6
12 36 24 36
12 100 24 100
) , $
1 .0
12 1 = 36 3
:& " D , . ! ) % % % !
2
6
6
3 10 4 10
4 10 6 10
7
36 100
$-
1 .0
A B
8
Repeated Trials
H I% < # = " 4
" (% (
$% $
$ 1 ! , . & ' 0 %
! < # 4 ! & $1 # E
.$1 # E / " / 0 1 % "
9 $ 6 9 - * 5 2 8 :+6
% % ? ' =
: /
% % ! N % $ - + 9 % (* )
( $!
=
< # = % ' =
: / "
$1 # E H I% 5 % . ? / * %
5 % . J ( $! N =
N P1) ( $!
1 = =
1 = ) ... ( $! = 25
N)
NP2)
" ; ' " 0 E % $ - % (* ) ' . $% ! $ N+ 1 $! 1 =
$!
!
$! 3
$! 4
$! 5
=
=
5
=
4
=
2
=
=
:& . $% !
3
1=
1
" " $% ! " B J
" , $# ! 4 & ' " D , . ) 4
.
& I% ) & % E% " )
B in o m il P r o b a b ilit L a w
:& O J
% % , . < 9 = $ 1 ! U C
½ = $ - < # = /
. " 3* ) $ - $ ( > - 3 < # = / /
½ . ½ . ½ = (½)3
! " 0 E $ - % > - 0 E % , ; '
< # . / : 1 % " - & ' & % % > - 3* ) < # =
> - 3* ) < # = /
! 5 , .(½)2 = J -
= " - 0 1 % & ' ( - < # . / (½)3 (½)2 26
< @ "
3* ) & ' = /
$ % & .
D,.
!
J & ' = / 4 % % % ! - & ' . / > 4 / % F
V - J & ' . / > - 3* )
/ J ! 4 # U C
.
> - 3* ) < # . / ! > - 3* ) J < # =
"
4 % ' E% 4 # < # = #
> - 3* ) $ ! ! 3 > - 0 E % ! ! & ! 3 > - 0 E % ! ! & '
4 % J '
( =
! D * D % J , ⊂ 53 J > - 3* ) $ ' !
& ' ! " ) J 3* ) ' D , . $ : ' ! < # = % ' 4 & ' '
4 C % 5 , $ !
⊂ 53 (½)3 (½)2
3 B - % $% $
= 1 0 (½)5 = 10 32
$ & ' : % % ' % > % O
(1-P) : - (P) 3 , . B - ! : -
3 (r ) & ' 3 , . B - ; ' (n) $
⊂ nr p
r
D , . % !
= ! ( n> r >
(1 -p )n-r
0)
& ' E ' % ! , !
& I% ) & % , . - = B - ' " ) " , $# " 1
27
2
= $ ! & ' " )
" 0 E % $ - & . % ) < 9 :( 13)
/ $= E " ; ' > - 0 E &
:& # ! % , . E ! :
= " 5 = /
⊂5 (1 2 )
(1 2 )0 = (1 2 )5 = 321
⊂ 4 (1 2 )
(1 2 )1 = 5 (1 2 )5 = 325
5
5
4
5
= " 4 = /
= " 3 = /
⊂3 ( 1 2 )
(1 2 )2 = 10 (1 2 )5 = 10 32
⊂ 2 (1 2 )
(1 2 )3 = 10 (1 2 )5 = 10 32
3
5
= = /
2
5
5 1
⊂
= = /
(1 2 )1 (1 2 )4 = 5 (1 2 )5 = 325
⊂0 (1 2 ) 5
0
= = /
(1 2 )5 = 1 (1 2 )5 = 321
28
& I% ) 5 ! 1 & ' $ & . H I% D , . / * %
(12 + 1 2)5
- n .
& ! 4 & ' 3 B - "
! 8 ; ' $ ' % $# 2 " & . n ... 2 1 1 =
, E?
P(O) + P(1) + P(2) + ... + P(n ) = 1
: 14
& % 4 3 $ / $% ! " =
." 3 %
$% ! " :
3 : / P D / P D / P
" 3* ) D / P
= ( 3 : / P
⊂
31 0
125 5 = 216 6 6 0
3
1 2 31 5 1
⊂
= D / P
1 25 75 = 3 = 6 6 6 36 216
29
⊂
31 2
15 5 = 6 6 216
⊂
31 3
2
1
= D / P
= " 3* ) D / P
1 5 = 216 6 6 3
0
$ ' % $# 2 " D , . @ 5 , = " B / * %
& I% ) 5 ! 1 & ' $ & . H I% D , . / * % !
1 5 + 6 6
3
- * 5 2 8 : +
" ! 4 $ & ' " % + & '
+ , . & ' " ) " , $# " & ' J / 0 1 %
B - ' V & ! 4 $ & ' " 0 % % E% " D , .
) & ' Q E < 9 $
3
. J
Hypregeometric Low
:& ) % , . U C % % % !
: 15
: & . # D > $# ) ! N : # E 0 ! % , E R .
" ! % + ! n3 + 3 0
! n2 + C ! n1
$ " ! ! 4 4 # 9 .+ ! r3 + ! r2 + C ! r1
"
.
N
⊂R
:
.
0 ! ! R E
, . n1 K + C " ! + C ! r1 4 3 $% ! n2 K + " ! +
! r2 4
n3 + " ! + ! r3 4
⊂r
2
n2 r2
n3
⊂r
(⊂ )(⊂ )(⊂ )
.
.
, . 1
n2
.
3 J n1 r1
n1
⊂r 3
.
, .
n3 r3
$ "
. , .
= ! &
n1
n2
n3
1
2
3
⊂r ⊂r ⊂r N ⊂R
3K K
R = r1 + r2 + r3 = n1 + n2 + n3
:
@ # ! n1 % $# ) ! N : # E 0 ! %
! , 9
R 9 % Z # ! nz ... & % ) # ! n2
3 1
# ! r1 $ " ! ! ; ' !
: . z # rz ... & % ) # r2 @ n1
n2
nz
1
2
z
⊂r ⊂r ... ⊂r N ⊂R
J & % < $V = D , . :16
% 20 $ I 2 : % " , E " 5 4 * 10 ' $-
: . '
.$ !
(i
." < # J (ii
:
= $ ! J % (i 10
< # $
5
⊂2 ⊂1 = 225 15 455 ⊂3
< # J & % "
4 * ( % ! S
10
< # J (ii
-% J , . -@ 5
⊂0 = 335 = 1 − ⊂3 15 455 ⊂3
" 3* ) $ < # J . 5
10
⊂1 ⊂2
+
5
10
5
10
⊂2 ⊂1 + ⊂3 ⊂0 15 ⊂3 3 2
=
335 455
Expectation
" $ / %
V & ' S 4 C
% F
V $# .
& C > -
E2 [ % < 9 P % C ' , ; ' $ I
= ; ' : % % : # = J , a # $ - < 9 X . V D , . : # S
E2 > - < P.X
4 ! × . = > -
E > - # %
& .
∴
E = P.X
i " % ! n > ' χ a # ! , 9 , .
= > - # $ $ ; ' I1
P.X (1 + i) − n =
P.X
(1 + i) n
: 17
$- 4 , 9 % % 5 4 * U . 4 $ & '
B % $ $- " % ! ( $- 52) 4 # $ Y 4 $ # ! + : ' 4 1
4
J , ) . ' "& " :
= "& " $- 4 = U
. > - $ - < # = % % ! I $ , . % C
, ; '
: ! )
E = P.X.
= 1 4 (5) = 1.25
. %
> %
J : ! J 3 3
:18
4 : ' / % ( . J ) & C > - a # !
Y 0. 4 0 4 ! ! , 9 %
100 D - . 4 ! & ' 4
40 × 100 = 40 = & C > - 100
E = P.X. =
5 > ' ) & ' ! , . $ - F
:
:19
1
= & C > - # $ $ ; ' % 3 . 5 I1 ! " %
E = =
= =
P.X
(1 + i) n 40
(1.035)5
40 1.1876862 3 3 .6 7 8 9 = 3 3 .7
< 9 . $ - > ' " & $ " , 9 : % , . < %
%
3 3 . 7 % ! - * $ % " ! 2 = I1
$! $ % I' > - ) " ! 2 D , .
$#
; ' " % 0 E
5 I' ! %
100 & . . $ - a # > ' $ ' ! ! ( % 3 . 5
." %
3 4
: V J , a # & . & C > - + 2 $ - " % !
& C > - ! ' ; ' . $ - . > - a # 4 ! $ ! & '
$ - > ' ? " ! 2 , 9 < # ? ! ' 0 D , .
$ > ' , ( % ? ) * ( : # N a # ) . .
? F
V & C > - # $
$! I1 0 < # & C > - # $ $ & . :2 0
$! 2 2 & ' S
E2 .
& ( % % 3 . 5
Y @ a # , 9 %
2000 a # : > ' ? " ?
. , E ! - < # + $ -
" # $ ! @ E " & & ' "
& $ 9 8 7
& . O <
( .S
:
& .
186 7 !
E2 100000 .
$ / % $ ' $ = - < # + l P = x+n lx
x % - < # -
& . 1x
x + n % - < # -
& . 1x + n
3
7 - & ! @
l 78106 P = 40 = = 0.84314 l 20 92637
3 5
& . & C > - # $ $
P.X
=
0.84314 × 2000 (1.035)
20
(1 + i) n
=
1686.28 = 8 4 7 .4 6 7 1.9897877
< # % N 2 & ' S
E2 ! < # , . < %
%
84 7. 4 6 7 : - % - ! 2 ? $! 2 < 9 > '
@ 5 , . @ a # , 9 %
2000 a # : > ' ? . 20 ( % % 3 . 5 -@ " ! 2 D , . . ) $! 2
( / ! %
2000 a # > ' & 1 !
! : % ; ' $% . @ < ! 2
4 0 < # " ) E & ' > - -
) a [ 5 , ) $ - E # $
.H % # > - 4 $ - E # $ B
> 2 & ' %
1000 a # ) & ' S
3 & .
E2
: 2 1
% D - ( ) ( & $ % =
% !
& '
E
U " :& # !
½ . J < # = P1
< @ $%
2 . J < # = P1 5
$ % ) $%
½ . %
( 1050) D - < # = P2
3 6
%
( 1102.5) D - < # = P2 3 . 5
' % 5 . $ @ & ' : I1 % # , ; ' % ( % # : I
Y ) +
9 ) +
S
E2 , . U = %
9 - , E :
:& # ! % $# ! 2 ) 0
1/ 2
2/ 5 3/ 5
10 0 0 1/ 2
1 05 0
2/ 5 3/ 5
0 1102. 5
0 1102. 5
4 : # = > - E # $ 4 % 5 ,
:& # ! 5 , ) a # $ - % [ % ) " # E 1. 0 – 10 0 0 2.
1102.5
(1.05) 2
=
− 1000 =
1050 − 1000 = 1.05 1102.5 1050 4 . + − 1000 = (1.05) 2 1.05
3 .
3 7
- 10 0 0 0 0 10 0 0
:& # ! 5 , " 4 % 5 ,
@ =
1 2 2 . = 2 5 10
& % ) =
1 3 3 . = 2 5 10
3 ) =
1 2 2 . = 2 5 10
> =
1 5 3 . = 2 3 10
" $ - E # E $ 4 % -
:& # ! 5 , > - E J > - 4 % % ! : % ' $#
1/ 2
0
3/ 5
10 0 0 1/ 2
2/ 5
1 05 0
2/ 5 3/ 5
0
1000 0
1102.5
2/10
200
3/10
0
0
0
2/10
0
1102.5
1000
3/10
300 100
$
% 100 : : # = > - E # > - $ - . ) +
3 8
; S
E2 , . U = % % % % ? ' , 9
3 ($- 52) 4# $ - " .1 1 4
:$ O " 4 .$ %) 4 - < @ $-
@ # - !
.
1 (J % ) ! 2 - ! .4 16 1 4
! 2 0 1 % - ! .R
( ' R ! % E . SE 2 @ R $) * ) % .2 1 4
3 8
: . ' (
0 % E $) * ) ' @ !
.
# % ! .4
4 $%! H I % = " 0 E % $ - " .3 > - $) * ) $ & ' =
$) * ) /
3 9
. % !
. .4 1 Y % 8
> C =
< # = , ; ' % $ - & SE 2
> C : %? ' $! < # = , 9 %= ) n $ # D, . SE 2 ! , ; '
.5
& ' + " ! 3 * )
.+ C ! + !
+ C ! D, . ! 4' . %= ! 4 1 6
! + " ! 3 * ) % $#) " ! 5 : @ ! .6
" ! > + " ! 3 * ) % " ! 7 : & %) C
% ! . ' 0 ! ! ! " .+ C (26/35)
Y + C -@ < #
+ " ! 3 + " ! 6 + C " ! 3 < # J 0 ! .7 . ' 9 ( I 2 " ! 3 " (1/10 )
Y ! %
4 + " ! 3 + C " ! 5 % $#) ! 12 : 0 ! .8 :$ O " 4 0 ! " ! 3 " + " !
(21/55)
(27/55)
+ " ! /
.
+ ! / .4
(34/55)
-@ < # + ! ! .R
(3/11)
) ! ! 5 %. 0 : % .D
(3/44)
#! " ! ! .
40
$- # % $- 10 0
! ' I 1 $- # % 50 . - I
"* 1 & ' > .9 "= = E
Y $-
! 30 + C ! 45 % # $#) ! 10 0 : 0 ! .10
0 ! < 9 " %
! " + ! 25 +
! " ) 0 ! < 9 " %
$ %) ! " )
< # $ 3 * ) " ! ! . $) ) (0 . 0 3375)
Y F
4
" ) ($- 52) 4# $ $- " .11 ( 1 / 2)
(1/52)
(10 /169)
:$ " 4 .$ %) $- "
< @ $- 0 1 % $ %) $- !
.
< @ $- 0 1 % & . $ %) $- ! .4
0 1 % - ! .R
(1/13) = 0 1 %
0 1 % - ! .
(3/169)
= 0 1 % - ! .D
$ " 3 % $ - " .12
+ ) * ) " & ' = & ' !
=
3 . ( 5 / 8)
3 . ( 1 / 4)
41
.
.4
3 * ) "
< 9 4. , , 9
2 : # 0 E ? ! , 9 .13 3
. ' . : 4. , , 9 1/6 $#' 4. , , 9 1/4 ( 2 : # E , 9 @ [ =
& ' : # E ? (13/36) Y $ I 2 $
% 8 $ % 5 < # J @ - %=
% ! , 9 .14
. ' .$ # 11 $ % 6 < # J & %) $ # Y ( - %= ( I 2 4 % ! (0 . 36878)
4 # 1 $ : #I E & . $ ! .15
4 ) $ D, & % L 1 4! . # 0 % ( 1 / 4)
(5/16)
(15/16)
:$ "
"% $#I %
% ) ! $#I %
) ! @ < # "% 3 $#I %
.
.4 .R
! , ; ' B , A = -% < 9 + % "! 2 Q 9 " .16
$= -% < # = 0 . 6 . A $= -% < # =
. ' 0 . 1 . ( = -% < # = 0 . 3 . B (0 . 8)
:
B $= -% A $= -% < # $! 2 = P 42
P ( AB) =
1 5
3 S & % L 1 & ' ) B , A B .17 P (B/A) =
3 2 , 5 5
1 2
P (A/B) =
1 3
P (B) , P (A) KK
M2 M1 "%! "%! , 9 $ I ! U = > %= & ' .18 R%M B 0 . 40 0 . 30 0 . 30 4 < # > %= M3
4 < # 3 * ) "% ! R%9 $I 2 3 1 ! , 9
: % @ R%9 ( I 2 = 4 .4 R%9 .
Y M1 $% ! > %=
[ = , . ! . ' 4
. (8/20 9/20 3/20 ) Y M3 > %=
Y M2 > %=
9 ($- 52) 4# $ "- 5 % , 9 .19 $- < # $E @ D, . + . ' .(3243/10 829) Y 4 2 =
2 $% ! $' " > %= R%9 % 60 H % 1 $% ! .20
R%9 % 2 % 4 1 $% ! R%9 % 5 .& - H % & ' "%=
- $ $ ! . ' .4 2 $% ! (30 /38)
43
Y 1 $% !
i) ii) ii)
.1
26 26 1 . = 52 52 4 13 13 1 . = 52 52 16
13 13 13 13 13 13 13 13 1 . + . . . = + + 52 52 52 52 52 52 52 52 4
& ' ! & . 23 = . $%! "
.2
3 J 0 %
$) * ) < # = .+ % 3
K K
1 1 1 + = 8 8 4
(1 / 2 . 1 / 2 . 1 / 2) + (1 / 2 . 1 / 2 . 1 / 2) =
44
1 4
(i
# < # = (ii 3 8
=
C2 3
8
45
=
=
3 = 8
25 = 32
5
C5 5 C4 5 C3 5 C2 5 C1 5 C0
÷ 32 = 1/32
$%! " .3
= $! " = = "
÷ 32 = 5/32 ÷ 32 = 10/32 ÷ 32 = 10/32 ÷ 32 = 5/32 ÷ 32 = 1/32
p ( S S S)
=
1 1 1 1 . . .= 2 2 2 8
.4
$ ! > " ! 3 > C : %@ 3n = $# $ % " !
.5 1/2 = $! < # = = =
< # =
"
= + " ! 3 ' > C & "
∴ n J + C ! ! ' > C & 2 n n ×1= 2 2
= + C " !
∴ = + " !
n n 5n ×3 + × 2 = 2 2 2
46
= + C $ ! ! n ÷ 3n 2 n 1 1 = × = 2 3n 6 =
:& . = $%! " .6 +
+ C
+ C
+
+
+
+ C
+ C
! ! S-% = + C -@ < # D ! !
.
9 26 3 3 1− . = 1= 5 7 35 35
+ $ %) + C < @ = + C -@ < # D ! ! + C $ %) + < @
+ C $ %) + C < @
2 3 3 4 2 4 = . + . + . 5 7 5 7 5 7 26 = 35
47
+ C " ! 3 * ) ! = 4 # .7 + 3 * ) + 3 * )
2 1 6 5 4 3 2 1 1 3 P = . . + . . . . + = 12 11 10 12 11 10 12 11 10 10 أو
=
C3 3
+
C3 12 C3 6
12
C3 i)
P
+
C3 3
$%! " .8 = 220 9
C3
( ) =
P (+ ! )
= =
iii)
P (+ -@ < # ! ) = =
48
3
12
C3
C0
84 21 = 220 55
=
ii)
1 10
=
3
C1
9
12
C3
C2
27 55
3
9
C1 C2 + C2 C1 + C3 C0 12 12 12 C3 C3 C3 3
34 55
9
3
9
= 1 -P (+ ! / ) 21 55
= 1 34 55
=
iv)
5
P ( # 0 1 % #! " ! ) =
C3
P ( # 0 1 % ! ) =
I $ - × =
=
1 100
4
3
C3 + C3 12 C3
3 44
=
v)
+
×
5
4
3
C1 . C1 . C1 12 C3
=
=
3 11 .9
50
1 2
. % = % ! 4
∴
. $- ∴ 4 # .10
P =
45 25 30 . . 100 100 100
= 0. 0337 5 49
< @ $- & ' # < # = 0 1 %
(i
1 = < @ $- 0 1 % 52
(ii
' % 5 %. @
1 /2 =
- ! = 4 #
10
.11
... 2 1
4 4 4 4 = . . + + ... 52 52 52 52 4 4 = 10 . 52 52 10 = 3 13 10 = 169 4 1 4 = 13 . = 52 52 13
8 $%! "
S =
(iii .i .12
5 $ "
5 = 4 # ∴ 8
5 0
P =
2 8
=
1 4
.ii
(1/3) = $) * ) " E ! .13
$#' 4. , E ? ( 2 4. , . 4 # P ( E ? )
E ? : 4. , E ?
1 2 3 3
11 3 4
1 3
1 6
= + + 13 36
=
1 5 × 2 13
1/2 = %=
J E .14
= 4 @ %= 4 - !
1 6 × = 4 & %) %= 4 - ! 2 17
P (4 ) =
1 5 1 6 + 2 13 2 17
= 0. 36 8 7 8
i)
P ( "%)
=
ii)
P ( % ) ! )
=
iii)
4
C1 16
4
C3
P ( ) ! @ < # "% 3 * ) ) =
1−
=
15 16
& I %) %-
.15
S = 24 = 1 6
+
16
4
C4
C44 16
4
4
4
4
C0 + C1 + C2 + C3
=
16
5 1
P (A B)
.16
= P(A) + P(B) - P(AB) = 0. 6 + 0. 3 – 0. 1 = 0. 8
.17 P(AB)
= P(A) P(B/A) = P(B) P(A/B)
∴
P(A)
=
P(AB) 1 / 5 2 = = P(B / A ) 1 / 2 5
P(B)
=
P(AB) 1 / 5 3 = = P(A / B) 1 / 3 5
! .18
M1 > %= M2 > %= M3 > %=
[ = ! = P(D)
[ = ! = P(M1) [ = ! = P(M2) [ = ! = P(M3) $ O " & $? "%
P(M1) = 0. 3
P(D/M1) = 0. 01
P(M2) = 0. 3
P(D/M2) = 0. 03
P(M3) = 0. 4
P(D/M3) = 0. 02 5 2
$ " 4% O % P(M1D) = P(M1) P(D/M1) = (0. 3) (0. 01 ) = 0. 003 P(M2D) = P(M2) P(D/M2) = (0. 3) (0. 03) = 0. 009 P(M3D) = P(M3) P(D/M3) = (0. 4) (0. 02) = 0. 008
. % # ) ! ( " D, .
$ / % E
P ( M1 / D ) =
=
P( M 2 / D) =
P(M1D) P ( M 1D ) + P ( M 2 D ) + P ( M 3 D )
0.003 3 = 0.003 + 0.009 + 0.008 20 P (N 2 D) P (M1D) + P (M 2 D) + P(M 3D)
=
0.009 9 = 0.020 20
P ( M 3 / D) =
0.008 8 = 0.020 20
5 3
:& $ " S E # !
9 /2 0
0 .0 0 9
0 .0 3
0 .3 0
M2
3 /2 0 8 /2 0
1.00
0 .0 0 3
0 .0 1
0 .0 0 8
0 .0 2
0.02 0
0 .3 0
M1
0 .4 0
M3
1.00
Q % E .19 P ( 4 2 ) =
=
4 1
c c 52 c5
4 48 4
3243 10829
& ' C #E 3 , . B - $ ' % " 0 E 5 %.
' / & < @ $ , E ? % 4 2 $- 4 4
= ! $ D, . & ' 4 4 2
4 48 47 46 45 4243 × × × × = 52 51 50 49 48 54145
. 4 # ' $#) 0 E " 3 3243 = 54145 3243 = 10829
5×
5 4
.20 P( D / M1) = 0 . 0 5 P( M) = 0 . 6
M1
P( M2) = 0 . 4
M2
D
P(M1D) = P(M1) P(D / M1)
= 0.60 × 0.05 = 0.03
N
P( D / M2) = 0 . 0 2
D
P(M 2D) = P(M 2 ) P(D / M2 )
= 0.40 × 0.02 = 0.008
N
P(M1 / D) =
P( M1D) P( M1D) + P( M 2 D) 2
0.03 0.03 + 0.008 0.03 30 = = 0.038 38 = 0.789 5 =
5 5
5 6
RANDOM
V ARI AB L E S AND P ROB AB I L I T Y DI S T RI B U T I ONS
& % . , ) 4 = 1 & ' %
H I % c ... 4 # $ - 4 D ! 4 % $ -
% $ & ' ! " % ! $ - $ I 2 4 # = ! &
" V < . 4 # $ # $ - " ! & ' ! $ % .$ I 2
(* =
DISCRETE ! ! & I 2 V
CO N TIN O U N S :
$ = @ - $ - , E ? J , & I 2 V .
... 3 2 1 )
: > V < # $ #)
D : % ! $ I = 9 $ % & ' @ '
P
c ... " " ! 2 Q M 2 & ' : "
P
Q * E & C 1 $ # . @
57
P
:
) Q E $ - J , E ? J , & I 2 V .
.$ 0
: = V < # $ # )
. @ " N Q M 4 # $ 2 "
P
.0 Q 9 & ' & I @ = 4 *
P
. $ $ #
.c ... E @ $ % * E & ! 1 % # ) D i s c r et e
P
P
P r o b a b ility D is tr ib u tio n s
! " $ % ! H I % ) % % % ! : % ; ' % D . % , 9
:& %
1
2
1 /6
3
1 /6
4
1 /6
1 /6
5
6
1 /6
1/ 6
% % ! : % ' . & B % ! % & . % , 9
:& % ! $ % ! H I % ) %
2
3
4
5
6
7
8
9
1 0
1 1
1 2
1 36
2 36
3 36
4 36
5 36
6 36
5 36
4 36
3 36
2 36
1 36
58
"
& I 2 V - C J , , .
: ; ' , ! . > & I 2 V # & > < / %
! " # # $ % & % ' ( ' ) # * + % " # . , - .
$ % , - #
Pr o b a b i l y De n s i t y Fu n c t i o n (P.D.F.)
$ 1 # E - , E ? X > V % ! , 9
xn ... x3 , x2 , x1 ,
: < , .
;'
i = 1, 2 ...n 3
P (X=xi)
. f(xi) $ ' ) ! $
x $ 0 1 - + ( 2 0 -
* - ( * x ! ( ! $
3 # ( * f(x) & , & i) f (x) ≥ 0
P(X=xi ) :
n
ii) ∑ f ( x i ) = 1 i =1
:& ! 2 X > V # $ ' ) ! : , E ? X = x
x1
x2 … … … … ..xn
f (x) = P (X= x)
f(x1)
f(x2) … … … ..f(xn)
59
: 1
= " . X > & I 2 V % ! , 9 5 5 K =
5 5 K =
K =
K =
}=
& % L 1 $ % ! "
2 1 0 & . X $ %! ,
f ( x ) = P (X = x)
f (0) = P (X = 0) = 1/4 f (1) = P (X = 1) = 1/2 f (2) = P(X = 2) = 1/4
:& # ! . $ % ! X & > ! X =
0
x
f (x) = P (X= x)
1
1 /4
2
1 /2
1 /4
:& # ! % , . !
1
{
: ; ' $ % $ - & % $ ! : < #
3/ 4 1/ 2 1/ 4 0 1
2
6 0
X
P r o b a b ility D is tr ib u tio n
$ & I 2 V , E ' % 4 V & ' % % !
- $ X V , E (C % % $ % ! $ -
& > : . % $ D , . $ % ! $ - $ - Q
. X V #
X ( ! $ X $ % & + ( - F(x) & , & F (x) = P(X ≤ x) =
P(X ≤ x)
∑
xi ≤ x
( *
* x 5 *
f ( xi)
C o n t i n u o u s P r o b a b i l i t y D is tr ib u tio n s
: % ? = & I 2 V & > % (C % % !
= V , $ 1 # E % J , $ I . D , . $ - ! S
% 0 ! ! : % ! d $ ! % ! $ W % F
E
: 2
1%
# E $ ! % ! & ' # E !
& = & I 2 V % . 0 ! ' E W < 9 0 !
: $ D , D + # 0 ! 100 &
6 1
! " # $ % & X 0 .9 0 .9 0 .9 1.0 1.0 1.0
( )
0
1 7 25 32 30 5 1 0 0
5 9
f (x)
0
1 5
0 .0 1 0 .0 7 0 .25 0 .32 0 .30 0 .0 5 1 .0 0
[ I 2 0 ! E $ - % % # '
& I 2 V , $ % ! ' 1. 05 0. 9 5 % - J
= V $ & ' $ # ! 2 H % % % ! 5 , . !
$ 1 '
! < #
.> V :
& ' . !
.$ # = $ I 2 " V : & ' " % % ! = E
! 2 & ' ! a ≤ x ≤ b J (a.b) Q & ' > (* = V X !
&
f (x ) p (c ≤ x ≤ d )
a
c
d
6 2
b
x
(a, d ) Q & ' x = & I 2 V > ' & ' @
:& # ! & P(c ≤ X ≤ d) :
f(x) $ = $ =
. ! 2 & ' $ # # / $ J x = d
P(c ≤ x ≤ d)
, x =c K
: ! , 9 X = V # $ ' ) ! $ f(x) !
i)
: f(x)
f(x) ≥ 0
∫ f (x) d x = 1
ii)
1 =
R
∫
b a
∫
∞
< % % " $ !
f (x) d x = 1 (a ≤ × ≤ b ) f (x) d x = 1 ( − ∞ ≤ × ≤ ∞ )
−∞
P( x ≤ a)
J a J - X ? & .
, ! . F (x)
F( x ) = P ( x ≤ a) = ∫ f (x) dx a
-∞
6 3
:& ! 2 & ' ! & .
f(x) f(x) F(a)
−∞
f(x)
X
x= a
:3
J ( a, b) Q & ' X >
b
P(a ≤ x ≤ b) = ∫ f ( x )dx a
=
b
∫
f ( x )dx −
−∞
a
∫
-∞
f ( x ) dx
= f (b) - f (a) : K
f(x)
F(b )-F(a)
−∞
f(x)
∞
a
b
6 4
x
& . : : & I 2 V f (x) = c
(5x )
x " ) c ! , 9 P1
x=0 , 1, … , 5
c " ) $ - K K
?'
" % ! $ % & ' 3 ) & I 2 V X ! , 9 P2 : $ = < # & . X : " ) C
X=x
0
f ( x) = P ( X=x)
C
1
2
3
4
2C 3C 4C 1.5C
5 0 .5C
C :
i) ii) iii)
P (x< 3)
: :
P (0 < x ≤ 4 )
P (0 < x < 2 )
X :
6 5
" % ! $ ! : < # = < $ % $ - " , 9 P3
< % $ - " & " ) & I 2 V X . X V : .$ ! : < # =
10 $ ( " ! ) : 2 E P4
? ' E 2 - ) X & I 2 V ! , ; ' : 2 . X K :
: $ ' ) ! & I 2 V X ! , 9 P5 f(x) =
i) ii) iii) iv )
1 2
0 ≤ x ≤ 2
P ( 0 .5 < x < 1.5 )
: K K
P ( x > 0 .25 )
P ( x < 0 .75)
P (x>3)
f ( x) =
2−x 2
0 < x < 2
6 6
" % ! , 9 P6
: K K
P ( 0 .5 < x < 1 )
i)
P ( x > 1.5 )
ii)
P ( x < 0 .3)
iii)
P ( 0 < x < 2)
iv )
! , 9 $ # I $ E ! $ 1 2 " % & ' P7 . 0. 6 . $ =
: 2 ' - $ =
: N
J < #
% 4
" 9 # 4 & $ # I @ %
. x $ M & '
. & x > 4 % , 9
K K
4 3 2 & . - $ ) * ) : # 4 P8
% C ' , 9 $ ) * ) - @ < # ! & ' 2 N 2 N 3
H % $ % ! H I % = . % - B ) x .& x >
1 . + = M & ' 4 U 4
= .
% ! , 9 P9
% ) x ? % C ' , 9 .4 * $ ) * ) *
. x K & > H % $ % ! H I %
6 7
P1
∑ f (x) = 1
a11x
∴
∑ C ( x5 ) = 1 C ∑ ( x5 ) = 1
C ( 1 + 5 + 10 + 10 + 5 + 1 ) = 1 C = 1/32
∑ f (x) = 1
: P2
a11x
i)
∴
C + 2C + 3C + 4C + 1.5C + 0 .5C = 1
∴
C = 1/ 12
P (x< 3) = P (x=0 ) + P (x =1) + P (x = 2) = C + 2C + 3C =6 C = 6 (1/ 12) = 1/ 2
ii)
P (0 < X ≤ 4) = P (X=1) + P (X=2) + P (X=3) +
P (X = 4) = 2C + 3C + 4C + 1.5C 6 8
:
= 10 .5. C = 10 .5 (1/ 12) = 10 .5/ 12
= 1 – [ (P (X=0 ) + P (X=5)] = 1 – (C + 0 .5 C) = 10 .5 C = 10 .5 12
iii)
P (0 < X <2) = P (X =1) =2C = 2/ 12 = 1/ 6 x : F(x) = P (X ≤ x) F(0 ) = P (X ≤ 0 )
= 1/ 12
F(1) = P (X ≤ 1)
= 3/ 12
F(2) = P (X ≤ 2)
= 6 / 12
F(3) = P (X ≤ 3)
= 10 / 12
F(4) = P (X ≤ 4)
= 23/ 24
F(5) = P (X ≤ 5)
=
6 9
1
P3 f(x) = (1/ 2)x x= 1, 2, …
X=x f ( x) = P ( X=x)
1
2
3… … …
… .10
1/ 10
1/ 10
1/ 10 … …
… .. 1/ 10
f(x) = 1/ 10
i)
P4
x = 1, 2, …
P (0 .5 < x < 1.5) =
∫
1.5.
1 / 2 dx
0.5
=
x 2
1.5 0.5
=½ ii)
P (x > 0 .25) =
∫
2
1 / 2 dx
0.25
= 7/ 8 iii)
P ( x < 0 .75) =
∫
0.75
1 / 2 dx
0.5
= 3/ 8
70
, 10
P5
iv )
∫
P (x > 3) =
∞ 3
=0
0 ≤x ≤ 2 1
i) P (0 5 < x < 1)
=
=
=
1 / 2 dx
∫
2−x dx 2
∫
1-
∫
x 0 dx -
0.5 1 0.5 1
x dx 2
0.5
= x
1 0.5
= 1/ 2 -
= =
iii) P (x < 0 .3) =
∫
0 .3 0
∫
0.5.
-
x2 1 4 0.5
3 5 = 16 16
∫
2
ii) P (x > 1.5)
1
2−x dx 2 1.5
1 16
2−x dx 2
= 0 .2775
71
x dx 2
P6
∫
2
iv ) P (0 < x <2) =
0
2−x dx 2
= 1
x
0
p(x)
0.4
P7
1
2
( 0.6 )
( 0.6 )
1
( 0.4)
3 ( 0.6 )
2
( 0.4)
( 0.4)
4 3
( 0.6 )
5 ( 0.6 )
4
5
( 0.4)
P8 x p(x)
4 1 9
5 2 9
6 3 9
7 2 9
8 1 9 P9
P (x) = Cnx p P (0 ) =
27 64
P (3) =
1 64
x
q
n-x
, P (1) =
72
27 64
, P (2) =
9 64
73
"1 =
% ( " ) "# F !
. F C : I = " % &I = M > 0 . > # > C ! . : "# D , .
. V * 0 . 2 % : "# D , . $ % % &# '
Expectation
' f(x) : $ ' ) ! &I 2 V x ! , 9
∫ x f(x) dx
∞
-∞
< # : % ! ( ! : % ) ( ) ! , . . > -
f(x) $ ' ) ! $ % C & " )
J $ D , . &' > - U = x V ! , 9
∑
all x
x P(x)
x V # $ $ P(x) 3
$ C $ # < 9 2 E 3
E(X) x > - %
: % C x V < # Q $ %
74
E (x) =
∫xf
∞
(x) dx = µ
-∞
=
∑ xP(x)
=µ
all x
.( * = x V ! 4
E & > # & E(x) > - < ! . x V # $ - $ # 4 V &' ( ) µ & M
: 2 % ( (* ) > V $ &' ) > - $ V =
/ *
4 > - ( i = 0 1 , 2) xi 1 % % $ # P(x) (%
. : % &. @ D , . ) !
&. ( ϕ (x) (* ) % ) x &' $ J ' ( I 2 ( V x ! , 9
:J : - &I 2 V Q E @
∑ ϕ (x) P(x)
E [ ϕ (x)] =
=
all x
∫ϕ
Range x
(x) f(x) dx
. (* = ( x ! 4 , K K E ?%
ϕ (x) = x r
E (xr) =
75
∑ xr
all x
P (x)
K K !
=
∫x
r
Rx
f (x) dx
%
r = 1 E (X) = =
∑x
all x
K K !
P (x) = µ
∫ x f (x) dx
Rx
=µ
: 1
&C > - . ' $ " 3 * ) $ % % $ - "
Y" = " / "
:& &' / % " &% L 1 ) % % % !
1 /8
S
S
S
5 S
3
1 /8
5 S
2
1 /8
2
1 /8
5 5 5
0
1 /8
S
5 5
S
S
1
1 /8
1
1 /8
2
76
S
5 5 S
1
1 /8
S
5 S
5
5
B X " % D =
: < # = " )
V # : " 3 * ) % : - X :& &' D X=x
0 1 8
f ( x) = P ( X = x) 3
∑
E(X) =
i=0
2
3 8
3 8
3 1 8
Xi P (xi)
1 8
=
1
3 8
3 8
1 8
0 ( ) + 1 ( ) + 2( ) + 3( ) 3 2
=
&% &' $ % $ $ ) % % % !
:&
f ( x ) = P ( X= x ) 3/8 2/8
• • • 0 1
2
77
3
X
S
P r o p e r t i e s o f E xp e c t e d V al u e
> - % E : % $ C $ # < 9 2
E , . , & &I = M V < # J ! E (ax) =
aE (X)
∑ ax P(x)
E (ax) =
:$
x a
:K K K .
all x
a ∑ X P(x)
=
all x
=
a E (X)
∫ a xf (x) dx
∞
E (ax) =
-∞
=
a
∫ f (x) dx
∞
-∞
=
a E (x)
! , 9 a ") X ; ' &I 2 V E (a) = a
78
E (a) =
∑ aP(x)
all x
= a =
=
: K K K .
∑ P (x)
a (1)
a
E (ax+ b ) = a E (x) + b
:$K K K
%
: K K K .
E(ax+ b ) = E (ax) + E(b ) = a E (x) + b : 2
E(2x + 3) = 2 E(x) + 3 # " $ % E [ E(x)] = E (x)
# " # " :K K K .
") J ") > - % !, ! $ ) $ - > - .> - J > - > - ,
' E(z) = z
> - < 9 % #' E(x) $ ) $ z
% E[ E(x)] = E (x) 79
&4E [ ( X - E ( x) ] = 0 E [ (x – E (x) =
: K K K .
E (x) - E(E (x)]
= E (x) - E (x) = 0 :
Y X
E ( X ± Y) = E ( x ) ± E ( Y)
: 1
E (X ± Y)
∫∫
∞
=
:K K .
(x ± y) f(x.y) d xdy
−∞
= = = =
∫ ∫ x f (x, y) dxdy ± ∫ ∫ y f (x, y) dxdy ∫ x [ ∫ f(x, y)dy]dx ± ∫ y [∫ f (x, y) dx]dy ∫ x f(x) dx ± ∫ y f(y) dy E(X) ± E(y )
: ! "
Y, X
E ( XY) = E ( X) E ( Y) E(X Y)
=
∫ ∫ X Y f (x, y) d x d y 8 0
: 2 : K K K .
∫ ∫ xy f (x) f(y) d x d y … … ∫ x f (x) d x ∫ y f (y) d y
= =
=
. *
E (X) E (Y) : 3
:&. : : I 2 V X ! , 9 f(x) = 2x
0≤X≤1
E(x+ 1)2 E(x)
E(x2)
∫x
f (x) dx
∫x
(2x) dx
1
=
0
1
=
0
∫ 2x 1
=
2
dx
0
= = =
2
x3 3
2
1 -0 3
1 0
2 3
8 1
K K ! K K E (x)
?' :
∫x 1
E(x ) = 2
2
0
∫
1
=
0
f (x) dx
x 3 (2x) dx
∫ 2x 1
=
3
dx
0
=
2
x4 4
1 0
1 2
=
E [ x2 + 2 × + 1]
E (x + 1)2 =
E(x2) + 2 E (x) + 1 = =
½ + 2 (2/ 3) + 1 = =
½ + 4 /3+ 1 17 6
Variance & Standard Deviation " K 2 # 0
K &K . J K K % K %
" K 2 # 0
! O 2 - % % % % ! J ! > (D is p e r s io n ) . &I 2 V & > #
K V &K > K ( &K ) : - $
K $ " # % & > ! < # % &I 2 8 2
% ! ) ! $ + 9 &' , 9 Q &'
: K &I 2 V - " 2 2 % Q " # J %
0
K K K ( 2 0
) ! < 9 % . % .Q E < 9 .J % &. &I 2 V # & > " 2
$ K 0 1 % K 0 &I 2 V & >
> % O % % . 1 J ! > $ &' : -
K "' % > 9 $ - $ "' % .J ! > $ % # &
-
X
X ! ! " " # $ % & µ E ( x ) =
' % ( X 2
σx
2
σx
= V a r (x ) =
) V a r ( x )
∑ (x − µ) 2 p (x)
all x
∫ (x − µ)
∞
=
-∞
2
f (x) dx
+, - ) # $ * = E [ ( x -u ) 2]
= E [ x - E ( x ) ]2
8 3
X
. - "' % > > - . J
- / * % X K $ &. 4 &' $ #E X , !.
- 0 1 % " 2 0 ' X < # = % &! $ % 9 - 0 1 % " 2 # 0 X . # 4
& , % % % ? '
.J % <
-
X % . * . X / σ x : σx
=
σ 2x =
E (x - µ ) 2
J K % 4 < 9 4 * U = % % .+
, . &' - &C - J ! > # : 4
B " K 3 * ) % : - + 9 . (1) ) <9 x & K .
$ K & K & . =
: / " & I 2 V
V # x ! 4 $ 1 ! σ 2x J K %
σx .
8 4
X = x
f ( x) = P ( X=x)
Xf ( x)
1
3/ 8
3/ 8
( 1-3/ 2) 2 = 1/ 4
1/ 8
3/ 8
( 3-3/ 2) = 9 / 4
0 2
1/ 8
0
3
3/ 8
6 /8 E (x ) 12 3 = 8 2
( x-E ( x) )
[ x-E ( x) ] 2 f ( x)
2
( 0-3/ 2) 2 = 9 / 4
9 / 32 3/ 32
( 2-3/ 2) = 1/ 4
3/ 32
2
9 / 32
2
σ 2x =
σ 2x =
J % σ x = = E
(x) =
µ
3 4
24 3 = 32 4
?' , ! .
3 4 3 2
X :
V a r (x) = σ 2X
σ2
=
E (x2) - [ E (x)] =
σ2
2
E (x2) - µ 2
: K K K .
= E [(x - E (x)]2
= E [(x2 – 2xE(x) + {E( x )}2 ] = E (x2) – 2 {E( x)}2 + {E( x )}2
= E (x2) - [E ( x )]2
85
: 5
K
K ' " 3 % : - & . ) <9
$ / % E x K J % 4 : & &
$
[ σ 2 = E (x2) - µ 2
X = x
f(x)
xf(x)
x2f (x)
0
1/ 8
0
(0)2 = (1/ 8 ) = 0
1
3/ 8
3/ 8
(1)2 (3/ 8 ) = 3/ 8
2
3/ 8
6 /8
(2)2 (3/ 8 ) = 12/ 8
3
1/ 8
1/ 8
(3)2 (1/ 8 )= 9 / 8 24 E(x2) = =3 8
E(x) =
σ2
3 2
J ' , ! .
= E(x2) - [E (x)]2 3 = 3 - ( )2 2 3 = 2
= # 4
& , J J % σ=
86
3 3 = 4 2
Properties of Variance
J @ 4 % # S
E D , . # S
:& . S
E D 5 % .
E . X & I 2 V & ' :
X a V ar (ax) = a2 V ar (x)
(1
: K K K .
V ar (ax) = E [ax – E (ax)]2 = E [ax – a E (x)]2 = a2E [x –E (x)]2 = a2v ar (x) : 6
:$ " V . ' 0 . 5 J x ! , 9 2X X 2 i)
V ar (2x) = 4 V ar (x) = 4 (0. 5 ) = 2
87
(i
(ii
:
ii)
X 1 V ar (x) V ar ( ) = 2 4 1 (0. 5 ) = 4 = 0. 125
' " ) a ! , 9
V ar (a) = 0
(2
: K K K .
V ar (a) = E [a –E (a)]2 = E (a – a)2 = E (02) = 0 a) = V ar (x) ± V ar (x
:$K K K
%
: K K K .
V ar (a) ± a) = V ar (x) ± V ar (x 0 ± = V ar (x) = V ar (x)
K V <9 : ) $ - J [ $ - J $ ' C 9 ' , ! .
. V , <# )N & I 2 88
: ! . ' 5 J x V ! , 9 : 7 i)
x+3
ii)
x–6
i)
V ar (x + 3) = V ar (x) = 5
ii)
V ar (x – 6 ) = V ar (x) = 5
:
: " # Y ! X (3 V ar (x+Y) = V ar (X) + V ar (Y) V ar (x - y) = V ar (x) + V ar (y)
B J % 1 # I 2 V B J . %
: K K K . V ar (X ± Y) = E [(X ± Y) – E (X ± Y)]2 = E [ {X − E ( x )} ± {Y - E(y)}]2 = E [ {X − E ( x )}2 + {Y - E(y)}2 ± 2{X − E ( x )}{Y − E( y )}] = E {X − E ( x )}2 + E{Y - E(y)}2 ± 2E [{X − E ( x )}{Y − E ( y )}] = V ar ( x ) + Var (Y) ± 2E [{X − E( x )}{Y − E( Y)}]
89
# ) 1 =
J E @ ? " )% % # ' , ! . ( E !
= 2 [ E {x − E( x )} E{Y - E(y)}] = 2 [(0) (0)] = 0
∴ V ar (X ± Y) = V ar (x) + V ar (Y) J : # " V
J <# D D , . ! :
V ar (x1 ± x2
± x3 ± …
± xn)=V ar (x1)+V ar (x2)+… + V ar (xn) C o variance
$ ' )! : f(xy) " % ! , ' Y, X V * 0 . : & # ! ( C
Co v (X, Y)
=
Y, X V V ' V , .
∫∫
∞
[X –E (x)] [Y – E(y)] f (xy) d xd y
−∞
= E [ {X − E( x )}{Y - E(y)} ]
9 0
P
C ) Y, X
! " # $ % $ ( Co v : & ' $
Co v ( x , Y) = E [ ( x - µ x) ( Y- µ y) ]
X
X, X 1
Co v (x, x) = V ar (x)
: K K K .
4 # <9 = % D * V : & ' Y X > C :
b ,a
Co v (ax, b y) = ab Co v (x, y) Co v (ax, b y) = E [{ax − E(ax)}{by − E(by )}]
2 : K K K .
= ab E {x − E( x )}{Y − E( y )} = ab c o v (x, y) Co v (x, a) = 0
9 1
a 3
Co v (x, a)
= E [{x − E( x )}{a − E(a )}]
: K K K .
= E [{x − E( x )}{0}] =0
P4
Co v (x1 + x2 , y) = Co v (x1, y) + Co v (x2, y)
: K K K .
Co v (x1 + x2 , y) = E [{( x1 + x 2 ) − E ( x1 + x 2 )}{Y - E (y)}] =E [{x1 − E( x1 )}{y − E( y )}] + E [{x 2 − E( x 2 )}{y − E ( y )}]
= Co v (x1 , y) + Co v (x2 , y)
x2 , x1 5
V ar (x1 + x2) = V ar (x1) + V ar (x2) + 2Co v (x1 , x2) V ar (x1 - x2) = V ar (x1) + V ar (x2) - 2Co v (x1 , x2)
: K K K .
V ar (x1 + x2) = E [ (x1 +x2)
- E (x1 +x2)]2
= E [{x1 − E( x1 )} + {x 2 − E( x 2 )}]2
= E [x1 − E( x1 )]2 + E[x 2 − E ( x 2 )]2 + 2 E [{x1 − E( x1 )} {x 2 − E ( x 2 )}]
= V ar (x1) + V ar (x2) + 2 Co v (x1 , x2)
: & ' J V 1 : & ' ) 9 2
V ar (x1 – x2)
x2 , x1 6
Co v (x1 , x2) = 0 Co v (x1, x2) = E [{x1 − E( x1 )} {x 2 − E ( x 2 )}]
: K K K .
= E[x1 x2 + E (x1) E(x2) – x1 E (x2) – x2 E(x1) = E(x1) E(x2)+E(x1) E(x2)–E(x1) E(x2)-E(x1) E(x2) = 0
! # V E (x1 x2) = E (x1) E (x2)
Cov (X , Y) = ρ Var (x) Var (y)
. 1 =
Y,X
J Y , X V ! , 9 1 =
, . !
Q & ' >
1 ≥ ρ ≥ -1 ρ2 ≤ 1 93
;K ' , K ! .
: " )? ' E (x) = µ ! , 9 (1
i) E (x- µ ) = 0 ii) E (x-c )2 = E (x - µ )2 + ( µ -c )2
5 4 3 K = E 2 S
. " ) J C 3
E 2 " " % ! , 9 (2
: 4 <# & . I # < @ $ & ' S
0. 04 0. 09 0. 12 0. 27 0. 43 0. 05
Y I # < @ : & ' . > - S
E 2 6
E 2 @
. '
a " ) ! % Q V = : % & ' ! E (x-a)2 " ) (3 . E(x) J
:& . $ ' )!
Y , X # V % ! , 9 (4
1≤ X ≤ 0
f (x) = 12 x 2 (1 – x)
1≤ Y ≤ 0
f (y) = 2 Y
Y
x
2
+
94
x Y
> -
$ - 6 : % 10 & : & I 2 V X ! , 9 (5 E (x2 + 3 x)
- $ "
& > & & (6 : 2 * E " & I
! ! "
0 10 20 30 40
0. 01 0. 05 0. 39 0. 45 0. 10 1.00
# $
. 2 * E : "
. 2 * E : "
: KK
(i
(ii
" * & ' : : % # " % 1 #
& P7 : % ! / % (
% & ' ! ! "
0 4 6 8 40
0. 13 0.27 0. 39 0. 21 0. 07 1.00
# $
95
$ @ "
: KK
(i
$ @ " # & (ii & K. < K @ I U > 4 = % $ - S
E 2 J 2 (8
0. 005 K K% 3000 & K. $ K % ) I % 4000 Y $ - D , 4 % . ' 4 < # 0. 008
C B A K E K K! , K' D + 2 S
E 2 4 . ,
2000 A K K ) ! 4 < # 0. 3 0. 3 0. 4
a K# . ' %
(9
3000 C % 25 00 B % YS
E 2 , . : ' > -
: K - " I C , B, A & . $ # > 2 $ ) * ) & ' $ ! 2 (10
4 K < K#
25 . 000 5 0. 000 100. 000 & K. $ ) * ) > 2 #
: . ' 4 < # % 3.000 5 000 10. 000 & . " % . $ ) * ) > 2 $ - " I &
. &
$ 1 =
(i
(ii
& ' $ + E @
& . X " % ! , 9 (11 f (0) = 0.9
f (1) = 0.05
f (2) = 0.03
f (3) = 0.02 96
Y $ 1=
200 & ' > - + E @
. '
: : & I 2 V X " ) C ! , 9 (12 f (x) = c x x = 3, 4, 5, 6 C " ) $ -
X > -
X
: KK
(i
(ii
(iii " ) (13
Co v (x, y) = E (x y) - µ x µ y & .
y, x V # $ ! 2 $ ' ) ! : " # , 9 (14 1≤ X ≤ 0
f (x, y) = x + y
1≤ X ≤ 0 Co v (x, y) J y, x V
(x, y) J y , x
97
KK
(i
(ii
(1 i)
E (x- µ ) = E[ (x-E (x)] = E (x) – E (x) = 0
ii)
E(x-C)2 = E (x2-2c x + C2)
@ P
= E (x2) – 2CE(x) + C2 = E (x2) – 2C µ + C2
@ P
E (x- µ )2 + ( µ -c )2 = E (x2 - 2 µ x + µ 2) + µ 2- 2 µ C + C2 = E (x)2 – 2 µ 2 + µ 2 + µ 2 - 2 µ C + C2 = E (x2) – 2 µ c + C2
4 # . ' ! ∴
X = x 1 2 3 4 5 6
f(x ) 0.05 0.43 0.27 0.12 0.09 0.04 ∑ XP( x ) E(X) =
allx
98
x f(x ) 0.05 0.86 0.81 0.48 0.45 0.25
(2
= 2.89
4 # . J : < @ : 2 " % ! , 9 Q V = : % & ' ! (3 1=
E ( x − a ) 2 = E ( x 2 ) − 2 a E (x) + a 2 144424443 Z
) , . 2 % Z K $ % (a
dz = 0-2 E (x) + 2a da = 0 – 2 a + 2a =0
a = E (x) 3
Q V = : % & ' ∴
Y X Y X E 2 + = E 2 + E Y Y X X
(4
1 1 = E (Y) E 2 + E(X )E Y X
D * : & ' H % F
99
* %@
% D < # > - ! % O
E (Y) = =
∫
1
∫
1
Y f (Y) d y
0
Y (2Y) d y
0
= 2 ∫ Y2 d Y 1 0
Y3 3
= 2 =
1 E 2 X
= =
1 0
2 3
∫
1
1
{12 X2 (1 - X) } d X
1
{12 X2 - 12 X3 ) } d x
X
0
∫
1
X
0
2
2
= 12 ∫ (1-X) d X 1 0
= 12
X-
1 = 12 2 = 6
10 0
X2 2
1 0
E (X)
∫
1
=
0
{
}
X 12 X 2 (1 - X) d X
= 12 ∫ X3 – X4 d x 1 0
= 12
X 4 X5 4 5
1 0
1 12 3 = 12 1 - = = 4 5 20 5 1 E Y
= =
∫
0
1 (2Y) d y Y
∫
2d y
1
1 0
= 2
Y X 1 1 E 2 + = E (Y) E + E(X) E Y X2 Y X 2 3 = (6) + (2) 3 5 = 5.2
10 1
(5 E (X2 + 3X) = E (X2) + 3 E (X)
V ar (x) = E (X2) - {E(X )}2 6
… … …
1
… … …
2
… … …
3
= E (X2) – (10)2
E(X2) = 106
1 3
E (X2 + 3 X) = 106 + 3(10) = 106 + 30 = 136
X–x 0 10 20 30 40 E(x)
=
(6 f ( x ) = P ( X= x )
Xf ( x )
X2 f ( x )
0.01 0.05 0.039 0.45 0.10
0 0.50 7 .80 13.50 4.00
0 5 156 405 160
∑ Xp( x )
E(x2) =
allx
= 25.8
∑ x 2 P( x )
allx
= 7 26
10 2
V (x) = E(x2) - {E ( x )}2 = 7 26 – 665.64 = 60.36 (7 X
P ( X)
XP ( X)
X2 P ( x )
2
0.13
0.26
0.52
6
0.32
1.9 2
11.52
0.7 0
7 .00
4
0.27
8
0.21
10 E(x)
1.08 1.68
0.07 =
4.32
∑ XP(x )
13.44
E(x2) =
= 5.64
∑ x 2 p( x )
= 36.80
V (x) = E(x2) - {E ( x )}2 = 36.80 – 31.809 6 = 4.9 9 04
≅ 5
10 3
E(x)
=
∑
allx
X P (X)
(8
= 4000 (0.005) + 3000 (0.008) = 20 + 24 = 44
E(x)
=
! (9
∑ XP( x )
allx
= 2000 (0.4) + 2500(0.3) + 3000(0.3) = 800 + 7 50 + 9 00 = 2450 " # $ % & ' ( ! (10 i) E ( X1 + X2 + X3) = E (X1) + E(X2) + (X3) = 100000 + 50000 + 25000 = 17 5000 ii) V (X1 + X2 + X3)
= V (X1) + V (X2) + V (X3) = 10000 + 5000 + 3000 = 18000
10 4
E (x)
=
= ) * * + , - . / 0 (11
∑
allx
X P (X)
= 0 (0.9 ) + 1 (0.05)+2 (0.03) + 3 (0.02) = 0 + 0.05 + 0.06 + 0.06 = 0.17
* +,
2 0 0 " 1 $ * +,
2 0 0 - . / 0 ' 2 * 3
= 200 (0.17 ) = 34
i)
∑
allx
(12
f (x) = 1
∴ ∑C X =1 C
∑X
=1
C (3 + 4 + 5 + 6) = 1 1 ∴ C= 18 ii)
E (x) =
∑ X P(X)
allx
=
∑X
=
1 ( 9 + 16 + 25 + 36) 18
CX
= C∑ x 2
10 5
=
43 9
iii) V (X) = E (X2) - {E ( x )}2 = E(X2) =
= C∑X
3
∑ X 2 P (X)
allx
1 (27 + 64 + 125 + 216) 18 216 = 9 =
216 43 V ar (x) = - 9 9 1944 − 1849 = 81 95 = 81
2
Co v (x, Y) = E (XY) - µ x µ y Co v (x, y)
= E [ (x - µ x ) (y - µ y )]
(13 :
= E (xy - x µ y - µ x y + µ x µ y )
= E (xy) - µ x µ y - µ x µ y + µ x µ y = E (XY) - µ x µ y
10 6
(14 Co v (x, y) = E(xy) – E(x) E(y)
. XY Y
X
∴
E(x+y) ! $ E (y) , E(x) ! " #
E (x + y)
= = =
= =
∫∫
1 0
∫∫
1 0
1
(x+y) f (x, y) d x d y x f (x, y) d x d y + 1
∫ [x ∫ f ( xy) dy] dx
014 04 42444 3 E( x )
∫ [x ∫ (x + y ) dy]
1
1
0
0
∫ x xy +
1 0
y2 2
1
1 = ∫ x x + dx 2 424 3 0 1 1
f (x)
1
0140442444 3 E ( y)
d y +
+ 1
=
7 7 + 12 12
1 0
1
0
0
∫y
1 0
x2 + yx 2
1
424 3 0 1 f (x )
y3 y 2 + + 3 4
10 7
∫ y [ ∫ ( x + y ) dx]
1
∫ y y + 2 dy
x = ∫ x 2 + dx + ∫ y 2 + 2 0 0 x3 x 2 = + 3 4
1
∫ y [ ∫ f ( xy) dx] dy
+
1
x f (xy) d x d y
0
1
d y +
0
∫∫
1
y dy 2
1 0
1 0
d y
d y
7 12 7 E (Y) = 12
E (x) =
1 2 1 = y+ 2
f (x)
= x+
f (y)
E (xy) = = = =
xy f (xy) d xd y
0
xy (x+y) d xd y
∫∫
x2y + xy2 d xd y
0 1 0
2 ∫ [∫ x y dx] dy +
∫ [∫
1 1
1 1
0 0
0 0
∫
1
x 3y 3
0
1 0
dy +
∫
1
0
∫
1 1 ydy + ∫ xdx 3 3 0
1 3
1
0
=
∫∫
∫∫
1
1
=
=
=
1
E(xy)
xy 2 dx] dy
xy 3 3
1 0
dy
1
∫
0
1 1 Y2 ydy + ∫ xdx = 30 3 2 1
1 1 1 + = 6 6 3 108
1 0
+
1 X2 3 2
1 0
C o v (x, y)
=
E (xy) – E (x) E (y) 1 7 7 + . 3 12 12 1 49 − 3 144 1 144
= = =
V (y) , v (x) ( ii V (x) =
E (x2) - {E ( x )}2
E (x2) =
∫
x 2f ( x ) dx
∫
1 x 2 ( x + ) dx 2
1
0
=
0
X 4 X3 + 4 6 = =
V a r (x) =
1
1 0
5 12
5 7 11 - = 12 12 144 2
109
V (y) = ρ( x , y ) =
11 144
Cov (x, y) − 1 / 144 − 1 / 144 1 == = 11 / 144 11 V ( x ) V(y) 11 11 . 144 144
110
111
THE DISCRETE PROBABILITY DISTRIBUTIONS
! " # $ ! " % & " ' ( ) *
! " # , ! " # $ + , ! " # $ + ,
! " -
. * $ # ! " ) %
$ & " . % / ! " $ + ,
. - 0 '
@ @The Binomial Distribution@
, - 0 & , $
2 ' ( ( * % $ & 3
3 , # + , & 3 4 & 3 5 . * , # 6
. 7 ! - % $ + % 8 - . 9
, + , + , & 3 ' ( $ 7 $ < . ; 3 . 9 100 :
: = 6 + ,
. .1
. .2
112
! " .3 .$
. #
% & ' % ( ) .4
! * + , - ' ) ! " .
/ ) % ( 0 " ( ! ! 0 " ! 0 1
. ! 0 " * +
:1
! 2 ( 0 " ( ! ! n 7 8
3 9 8 ! P ( % 3 ( % 4 5 6
3 & = / ! q + & + (1-P) ( ; ) : ! 3 3 & = ! ; ( ; ) 7 8
n = 1 :
( ( ?
S = (@ A 7 8
7 8
)
3 9 6 + A
3 3 9 6 + @
C
3 & = ! 0 % 1 " X D
1 E 0 ! X 2 ( /
7 8 & = ! P
( & = ! q= (I-P)
:! 9 x 1 ! * + 113
1
n=
1
X = x
f (x ) = P (X = x ) f (0) =
0
P ( ) = q
1
f (1) = P () = P
q+ P = 1
n = 2 :
( D
S = (@ @ E A F 8
@ E @ A
( " F 8
E A
9 G ! % A
A
)
A
( ( " ( 9 G ! % @ @
C
( ( " F 8
F 8
9 G ! % @ A
( " ( 9 G ! % A
@
4 ( ;
! 7 H
qq , qp, pq, pp
( ) C
2 E 1 E 0 ! ( x / 4 114
:! ! x 2 ! * + (2)
n= 2
X = x 0 1
f(x ) = P (X = x ) f ( 0) =
P ( ) = P () P () = q q = q
f ( 1) = P ( ) + P ( ) = P () P () + P () P () = P q + q P
2
2
= 2q P
f ( 2) = P ( ) + P () P () = P P = P
q2 +
2 qP + P
2
=
( q+ P )
2
2
=
1
3 n :
( ( ?
S=
{
}
8 = ( D ;
! 7 H & F = qqq, qqP, qPq, qPP, Pqq, PqP, PPq, PPP
3 E 2 E 1 E 0 ! ( x / 4
:! ! x 2 ! * + 115
3
n= 3
X = x
f(x ) = P (X = x )
0
P ( ) = P () P () P () =
f ( 0) =
q
3
f ( 1) = P ( ) + P ( ) + P ( )
1
= q
2
p + q
2
+ q p
2
p + q
p = 3 q
2
+ q p 2= 3 q p
2
2
p
f ( 2) = P ( ) + P ( ) + P ( )
2
= q p
2
3
f ( 3 ) = P ( ) =
q 3 + 3 q 2p +
P () P () P () = p
3 qp
2
+
P
3
=
3
( q+ p ) 3 = 1
: ( ( ) ( " . " = .
2= 4= 8=
21 =
22 =
23 =
( 1 ; ) n H 6
( 2 ; ) n H 6
( 3 ; ) n H 6
2 ( 2 0 " ( ( ( 7 H / %
! 2 2 ) H 6 . n 3 ( H 6 2n ; ) I 2 , 2 * 2 ) ( F 8 n F +
2 ) ( ) % ( ) ( " . " ! ( .@ H & ) ( ( 3 ( 116
F x F 8
3 9 8 J * ) #
(3 2 ( % 4 5 ) n ( 0 " ( ! ' D 2 4 2 ( ( % ( F 8 K
! @ H F (n-x) ( 4 / F x (F 8
3 & = ) C .(
Px F x (F 8 ) C D 4 qn-x F
n-x C D 4 /
px F x C D 4 K '
F 4 qn-x
; ) , 3 ) C D 4 (
C F x 7 8
E 0 7 8
n x
n n! = = x x! (n - x) !
; n x ,
3 9 8 K H
(nx )p
x
: 9 6 L ) n '
qn-x
3 & = 3 ; ) n E L. 2 2 "
: ! 9 ! 3 E 2 E 1
(33) p3 , (23) p2 q , (13) pq2 , (03) q3 p3 , 3p2q ,
3pq2
, q3
(3) ! & 8 ! I ! 117
(nx )p
( / = .
qn-x
x
x = 0, 1, …
.., n
x
(q+ p)n (q+ p)n = qn +
(1n )pq
(n2 )p q (nn−1 )p q +
n-1
+
2 n-2
n-1
+ … +
pn
(nx )p q
x n-x
+ …
+
! 0 " * + ! * + H ( ) 5 : ! 0 " * + 8 :
% x ! " # $
* " + , n & ' ( " ) # - + % $
(nx ) p
f ( x) =
x
qn-x
x= 0, 1, …, n
" + , 0 & ' $ p
" + , 0 & ! 1 $ q p +
118
q= 1
(i
(ii
(iii (iv
:.
: 2
! * ' ( ) ( ) ! ) ! K / % H 6 .( ) ) ) G ( 8 $
% ( ) ( M
: % ( M ) ) % 10 ) ; 9 ;
) / & "
) / & " "
(i
(ii
(iii
( 1 7 H ! ) " x D : 20 E ... E 2 E 1 E 0 ! x 2 ( / ) ; ) p D
) ; ) q ∴
i)
p (
p=
0. 10
q=
1-P
n=
20
! " # # $ ) = f (0)
f (x) =
f (0) = =
(nx )p q (200 ) p q x
= 0. 9 0
n-x
0
20
(q)20 = (0. 9 0)20
= 0. 122 119
ii)
P ( ) = f (2) f (2)
(202 ) p
=
=
2
q18
19 0 (0. 01)2 (0. 9 0)18
= 19 0 (0. 01) (0. 9 )18 = 0. 28 5 iii)
P ( ) =
≤ 2)
= = = =
p (x > 2) = 1- p (x 1 - {p (x = 0) + p (x = 1) + p (x = 2 )}
{
}
1 - (0.9) 20 + 20 (0.1) (0.9)19 + 0.285
1 - { 0.122 + 0.270 + 0.285} 0. 323
! 0 " * + * 4
F 2 n ( 2 ( % 4 5 6 ( , ) " 9 6 F %
x ! 0 % 1 ! * + 1 ; ) n % * 2 + H & * 4 (1) ! 9 % F 8
3 & =
2 2 2 2 2 2 2 2 2 ! 2 2 2 2 2 2 2 2 2 K 2 2 2 2 2 2 2 2 2 2 )
:! (4 )
12 0
(4)
n= 1
X 0
f (x ) q
x f (x ) 0
1
p
p
1
µ = ∑ xf ( x ) = p
0
x ! 0 % 1 ! * + ? 2 ; ) n %
2 ) ) * 2 4 (2) ! 9 % 7 8
3 & =
:! (5) ! K ) * + H & (! ) (5)
n= 2
X 0 1 2
f(x )
q 2pq p2 2
x f (x ) 0 2qp 2p2
E ( x )= 2 p ( q + p ) = 2 p
x ! 0 % 1 ! * + ? 3 ; ) n %
) * 2 4 (3) 2 ! 2 9 % F 8
3 & =
:! (6) ! 3 ) * + H & (! ) ) 12 1
(6)
n= 3
X
f(x )
Xf ( x )
1
3p2q
3q2 p
3
P3
0
q3
2
0
3qp2
6 q p2 3p3
µ = E ( x )= 3 p ( q
2
+ 2 q p + p 2)
= 3 p ( q + p )2 = 3 p
! 0 " * + (* 4 ) ! ) ) J = . H 1 ; ) n H 6 P
:
2 ; ) n H 6 2p 3 ; ) n H 6 3p
! 0 " * + * 4 J ( 7 H / % #
. n 2 n p ; )
q p #$ n % ! "
#$
µ = np
12 2
! * 4 % / ) L ' @ H µ = E (x)
n
∑ x f (x)
=
x =0 n
∑ x (nx ) p x q n - x
=
x =0 n
x n! px qn-x x ! (n x) ! x =0
∑
=
n
n! p x q (n -1) - (x -1) (x - 1) ! [(n - 1) - (x - 1)] ! x =1
∑
=
(n - 1) ! p x -1 q (n -1) - (x -1) (x - 1) ! [(n - 1) - (x - 1)] ! x =1 n
np ∑
=
n
np ∑ ( nx -−11 ) P x -1 q (n -1)- (x -1)
= =
x =1
=
np (q+ p)n-1 np
( 2 % 4 F 100 ( ( % 4 H 6 : 3 & O F 8
3 & =
3 2 2 & = ! 0 % 1 " x D : & & 2 ! 0 " * + & x ? H F 8 P=
1 2
n = 100 12 3
np = ! 0 " * + * 4
1 µ = E (x) = np = 100 = 5 0 2
K 2 2 2
# o
/ ) F n ( % 4 5 6 ( (1) " 9 6 F %
3 E 2 E 1 / 2 H 2 J n 2 % σ2x = ∑ (x - µ) 2 f(x) 4 . ! 0 " * + ( ( 1 8
9 6 8 * )
x ! 0 2 % 1 ! * + 1 ; ) n %
H 2 * 4 (1) ! 9 % F 8
3 & =
2 * + H p (4) ! K ) * + :! (7) ! 3 )
(7)
n = 1
X 0 1
f(x ) q p
x -µ 0-p 1-p
124
σ 2x =
( x - µ )2 f ( x ) q2 q (1-p)2 p p 2q
+
q 2p
= pq ( p+ q ) = pq
x ! 0 2 % 1 ! * + ? 2 ; ) n %
H 2 * 4
(2 ) ! 9 % F 8
3 & =
2 * + H (5) ! K )
2p * +
:! (8 ) ! 3 6 8
(8)
n=2
f(x )
x –
0
q2
-2p
4 p2 q2
1
2pq
1-2p
(1-2p)2 2pq
2
p2
2-2p
4 (1-p)2 p2
(&
( x -u)
2
X
u
f(x )
σ 2x = 4
2 2 p q+
( 1 -2 p) 2 2 pq + 4 ( 1 -p) 2 p2
= 4
2 2 p q+
( 1 -4 p+ 4 p2) 2 pq + 4 q
= 8
2 2 p q+
2 pq - 8 p2q + 8 p2 q
= 8 p2 q ( q -1 + p) + 2 p q = 8 p2 q ( 0 ) + 2 p q = 2 pq
125
2
p2
x ! 0 2 % 1 ! * + 3 ; ) n %
H 2 * 2 4 (3) ! 9 % F 8
3 & =
( ) ( I 3p (6) ! K ) * + 3 ; ) n 3pq ; ) * + H 9 6 8 :; ) ! 0 " * + H 1 ; ) n
pq
2 ; ) n 2pq
3 ; ) n 3P q
; 2 ) ! 0 " * + ( 7 H / %
n 2 npq
#
q p
# $ n & ) # "
#$
! * 4 4 / ) ' @ H 3
σ 2x = E (x2) – [ E (x)] E (x2) = =
2
n
∑ x 2 f (x)
x =0 n
∑ x 2 ( nx ) p x q n - x
x =0
126
x2 n ! px qn-x = ∑ x ! (n - x) ! x =0 n
= = =
n
∑
x =0
x(x-1)+ x 2 x2 $
[x(x - 1) + x] n ! x n − x p q x ! (n - x) !
%
n
n x(x - 1) n ! x n − x xn ! p q + ∑ x ! (n - x) ! ∑ x ! (n - x) ! p x q n − x x =0 x −0 n
∑
x =2
n(n - 1) (n - 2) ! px q( n-2) -( x-2) + E (x) (x - 2) ![(n - 2) - (x - 2)] !
= n (n-1) p2
n
∑
x =2
(n - 2) ! px-2 q( n-2) -( x-2) + np (x - 2) ![(n - 2) - (x - 2)] !
= n (n-1) p2 (q + p) n − 2 + np = n2 p2 – np2 + np
σ 2x
= E (x2) – [ E (x)]
2
= n2 p2 – np2 + np – n2 p2 = np – np2 = np (1 – p) = npq
σ 2 = npq H
! 2 % H 2 ; 2 ) 2 ! 0 " * + ; % σ =
σ2 =
127
K
npq
: 4
7 10 0 ( % 4 5 6 3 " 9 6 F % O F 8
3 & = % ; %
:
3 & = ! 0 % 1 " x D 1 2 1 q = 2 P
3 ! 0 " * + & x H 7 8
=
n = 100
= σ 2x = npq ; %
1 1 = 100 = 25 2 2 =σ =
npq =
25 = 5
D 2 4 2 & ! K ! ) * + / ) F ( + 7 . L 8
/ * ' C
F 2 ( 7 9 (P ... / ( ) ( 4 " )
2 ! 0 " * + / ) E (( ) " ) + 128
! / * ' C D 4 ( %
: ) * + ( " .
: " + 3 = 9 6 Q ! 0 ) ( . ( D ) ! ) C
R
.( ) ! ( 1 !
R
.( 4 ! ! ) ! ( &
R
./ ! ( + ! %
R
( :
( ) F S 4 ( % 4 ! F K %
3 & @ ) ! F K %
T 2 6 2 & ! 2 ( 8 ! ( % 5 G
)
R
R
R
/ 2 2 ( 4 H * 1 ! ) 1 H
( 2 + 2 ! 2 ( 8 ( + F ! ... 2 1 8
:( ( " . " , K ( 8
+ ! ( ) R1 F 1 8
.+ 8
( + F 8 4 F . R2
.+ H ( F 7 H * K ) +
129
( F 8 4 ( + F . " R3 .3 6 C L L 1 8
13 0
F 1 8
+
* 2 + ! 2 ! 0 % ) 1 ! * +
:! % K ) ! % )
X ! "
X
' ( ) X % & ... # 2 # 1 # 0
: * ' + .
f(x ) =
e−λ λx x!
... # 2 # 1 # 0 =
2. 7 18 28 . / =
x
,
e
+ " 4 0 1 & 2 3 ) ) ( + = λ . 5 ' 0 ) )
! " # $% # & ' ( # ) # *
:5 (
/ # # + , % - # " .
. / ' " % # $ % 0 #
13 1
:
! H 0 + ! 0 % 1 " x D / 2 ) . 3 ; ) λ ( ) ! ) * + f (x) =
e−λ λx x!
x 2 ( 4
: ) * + ( 1 8
e−3 (3) 4 4! 0.05 (81) = 24
p(x=4 ) = f (4 ) =
=
135 800
(e-3 =
0 .0 5 C )
= 0.16875
:6
F ( + ( ! K % ) H 6
: / 4 10 × 3 ( * / 4 20 .K 9 ;
. 4 G 9 K 9 ;
13 2
(i
(i i
:
10 × 3 ( ! K % " x D
! 3 % ! ) * + x 2 λ =
1. 5
10 × 3 ( ! K % ) ; e−λ λx f (x) = x!
i) P (x=0) = f (0) = 0. 223
3 × 10 = 1.5 20
: ) ( 1 8
,
e−1.5 (1.5)0 = e-1. 5 = 0!
ii) p ( 4 G 9 K ) = 1-p (x=0) = 1 - 0. 223
= 0. 7 7 7
/
) * + & ! ! ) 1 " x H 6
e−λ λx f (x) = x!
x = 0, 1, 2, … λ >
0
; * + ( % ; ) x ! 0 % ) 1 * 4
: ; λ ; )
13 3
µ x = E (x) = λ σ 2x = E (x ) – [ E (x)] = λ 2
E (x) =
2
∑x ∞
f (x)
x =0
∞
=
∑
=
∑
0
∞ 1
: 2 2
x
e-λ λx x!
e - λ λx ( x − 1) ! ∞
= λ e-λ ∑ 1
λx -1 ( x − 1) !
' ! %
eλ = 1 +
1 λx λ λ2 + + ... = ∑ 1! 2! x! x =0
∴ E (x) = λ e-λ eλ = λ E (x2) = =
∑ ∞
x =0
∞
∑ 0
x 2f ( x )
x 2 e-λ λx x!
W X(X-1)+ X ( X2 $
13 4
%
= = =
∞
∑ 0
[x (x - 1) + x] e-λ λx x!
∞ [x (x - 1) e - λ λx
∑
x!
0
∞
∑ 2
∞
+∑ 0
x e-λ λx + E (x) ( x − 2) ! ∞
= λ2 e −λ ∑ 2
λx - 2 +λ ( x − 2) !
= λ 2 e- λ e λ + λ = λ2 + λ v a r (x) = E (x2) – [ E (x) ] 2 = (λ2 + λ) - λ2 =λ
135
x e - λ λx x!
- 8 4 ) E ( 5 & 0 & L 8 20 3 , 8 (27/64)
: K ) F X * ( 0 (
(1
( ) K % 8 9 8 (i
(175/256) ( ) K % 4 G 9 8 9 8 (ii K K ( 0 ) 10 H 6 (2 :( 0. 2 ! 0 Z ) ; 9
(0. 1074)
(0. 8926)
Z ) ; 9 K K (i " Z ) 9 K
(0. 2)
(ii
( 0 ) 3 9 K (iii
% 3 + T 6 Q 6 0. 15 H 6 (3 ( 2 + & H ? . 5 . % 9 6 + & 15 E : ( )
K % + ; 3 (0. 0873)
(0. 3185)
(i
% 4 (ii
136
& = " k 4 H 6 (4
. k 2 ! * + * G 7 H ! ( ) 3
; 2 ) 2 H ? 5 & 7 ) (5 : E 1 /2 ; )
(1/16)
I I G *
(i
F 4 G 9 F ) G T (ii
(15/16)
! 2 % ! 4 2 L ! % & " " H (
: K ) J ( % 7 H H 7 + H ? E L '
(128/2187)
(448/2187) (1/2187)
(6
5 7 ; 9 8 (i
5 7 7 9 8 (ii
5 ' 7 ; 9 8 (iii
1 4 ) ! H 6 (7 ( 2 ) 7 2 ( ( ) ( ! ( ( !
2 2 4 @ H ! ( 0 / 4 ( % ) 7 ( " (0. 075)
(0. 225)
(0. 700)
:
( 1 M ( & *
(i
1 (ii
( 1 ( & " " (iii 137
2 & K ) 6 I ( % 4 & 9 8 (8 . 8 % I 9 L. 8 (63/256)
1/5 ( ! ( ) 7 K . & H 6 (9 .3 25 4 G 9 4 9 8 (0.766)
3 Z ) 9 K ( ! (10
2 I 2 Z ) 1/3 ( 8 .(16/243) ( 8
( 6 3 K Z )
6 ( 0 3 E ( 0. 30 $
(0. 58) O 4 G 9 /& " E A
(11
. 0. 90 ! 2 2 2 2 + + ! 2 ( 2 2 " ( 2 2 H 6 (12 D 4 E + + 10 3 2
(0. 6513)
3 9 3 ) ; H ! 0 " * + p , n % / 4 (13 ( n = 15 E p =
138
3/5) 18/5
E
p =
3 ; H ! 0 " * + * 4 (14 0. 2
(µ = 7
σ2 = 5 . 6 )
3 + ! % J 232 3 E ( 8 : ( 8
e-1/2 = )
(2e-1 = 0. 7 35 6 )
n = 35
232 Z K (15
K ) 0
J ' &
(i
J 4 &
(ii
(0. 18 39
C 4 * 8 ! % C ) H 6 (16 (e-4 = 0. 018 )
(0. 426)
: &
% & ! C ; *
(i
% & ! " G 9 C ( " . " * (ii
" ( + ! % ) H 6 (17 E ) * + * % H 6 , 0 4 I
:
139
(e-2 = 0. 135 ) , 0 4 I . * ; 8
(0. 382) , 0 4 10 . 8 % 4 "
(i
(ii
. ( * ) 9 (18 ) 8 ! ( & ) H 6 4
(0. 09)
O 4 !
T 6 & ! ( % 5 G ) H 6 (19 : E ( 8 ! 5 4 ) 5 3 8
(e-4=0. 018 ) (0. 91)
*
8 ( ! 4G 9 J K
(i
(ii
7 ( + 3 ! K % ) H 6 (20 R:/ 4 10 × 6 3 ( % / 4 F
(e-6=0. 0025 )
(0. 997 5 )
K )
K 4 G 9 & )
(i
(ii
( ( G ! [ C ) 3 Q X * 8 ! (21 ! K ) 200 [ ) L 0 . 500 3 " G 9
(0. 5 4 3)
(i
[ C . " 4 G 9 (ii
(0. 4 5 7 )
14 0
% / K 8 , 0 4 3 1 ! K C (22
. /7 × /3 % 3 4 E 2/2 ! (0. 00001)
K % 3
(0. 0001266)
(i
K " G 9 & (ii
14 1
@
( ) ( - 8 9 x 1 D (1 . 4 E 3 E 2 E 1 E 0 ! ( x / 4
p=
5 1 = H 0 8 K ) " p D 20 4
F X / K ) @ H ! 0 " x 1
9 6 1 8 K )
1 ; ) " 4
n= 4
(i
x= 1
1 4 3 q= 4
p=
: 9 8 ! 0 " * + ( 1 8
4 1 3 P (x =1) = f (1) = 1 4 4 1
3
4! 1 3 = ! ! 1 3 4 4 1 27 = 4
4 64 27 = 64
14 2
3
/ )
(ii P (x ≥ 1 ) = p (x =1) + p (x =2) + p (x = 3) + p (x = 4 ) = 1 – p (x =0)
4 1 3 = 1 – 0 4 4
0
3 = 1 - 4
4
4
= 1 – 8 1/ 25 6 = 17 5 / 25 6 x / 4 K & K ! ( 0 ) G " x D (2
0 E 1 E … E 10
! ( H 2
n = 10 p = 0. 2 q = 0. 8
i)
f (x) =
( ) px qn-x n x
p(x = 0) = f (0) =
! 0 " * + 3 1 8
( ) (0. 2)0 (0. 8 10 0
= (0. 8 )10 = 0. 107 4
14 3
)10
/ )
ii)
p(x ≥ 1) = f (1) + f(2) … +
f(10)
= 1 – f(0) = 1 – (0. 8 )10 = 0. 8 9 26 iii)
p (x = 3) = f(3) =
( ) (0.2) 10 3
3
(0.8) 7
= 120 (0. 008 ) (0. 8 )7 = 0. 2013 ≅ 0. 2
x / 4 . % 3 3 % F + & G " xD ( 3
0 E 1 E 2 E ... E 15 ! (
% + &
p = 0. 15
) + &
i) p (x = 0) = f(0) =
( ) (0. 15 15 0
q = 0. 85
n = 15
)0 (0. 8 5 )15
= (0. 8 5 )15 = 0. 08 7 3 ii) p ( x ≤ 1) = p (x = 0) + p (x =1) = f (0) + f(1) 14 4
=
( ) (0. 15 15 0
( )
1 14 )0 (0. 8 5 )15 + 15 1 (0. 15 ) (0. 8 5 )
= 0. 08 7 3 + 15 (0. 15 ) (0. 8 5 )14 = 0. 08 7 3 + 0. 2312 = 0. 318 5
: ! k 2 ( / (4 0E 1E 2E 3E 4
3 ) 3 9 8
n= 4
p=
q=
1 6 5 6
4 1 5 p (K = k) = f (k) = K 6 6 k
4−k
K 1 ! * + ! % !
4 K
Pk
Pk
0
(5 / 6)4
0. 48 23
1
4(1/ 6) (5 / 6)3
0. 38 5 8
2
6(1/ 6)2 (5 / 6)2
0. 115 7
3
4(1/ 6)3 (5 / 6)
0. 015 4
4
(1/ 6)4
0. 0008
145
! ( x / 4 G G " x D (5 0 E 1 E ... E 5
i)
p=
n= 5 1
2
q = 12
4G 9 7 7 ) G T
; E ( % 3 ". " " & . 5 5 &
; ) K
P = P(1) + P(2) + P(3) + P(4) =
=
( ) (1 2 )(1 2 ) + ( ) (1 2 ) (1 2 ) + ( ) (1 2 ) (1 2 ) + ( ) (1 2 ) (1 2 ) 4
5 1
5 2
2
3
5 10 10 5 15 + + + = 32 32 32 32 16
5 3
3
2
5 4
1
4
( ) + ( ) ) R1 = K (
= 1 - [P (5) + P (0)]
1 1 15 = 1- + =
ii)
32
32 16
/ & + / & ; ) K 146
P = P (0) + P (5 ) =
1 1 1 + = 32 32 16
7 D + & 8 ! 5 " x D (6 7 .... E 2 E 1 E 0 ! ( x / 4 E H 5 7 9 8 p D
5 7 9 8 / q p=
7 1 2 i) p (0) = f (0) = 0 3 3 0
q=
1 3 2 3
7
128 2 = = = 0.05853 3 2187 7
7 1 2 ii) p (1) = f (1) =
1 3 3
6
1 2 448 = 7 = = 0.2048 6
3 3
2187
7 1 2 iii) p (7 ) = f (7 ) = 7 3 3 7
0
1 1 = = = 0.0005 3 2187 7
E 0 ! ( x / 4 E 3 1 & " x D (7 9 ... E 2 E 1
147
n= 9
p= q=
1 4 3 4
3 1 M ( & * (i L 1 & 8
0 9 1 3 9 P (x = 0) = f (0) = 0 4 4 9 3 = 4
= 0. 07 5
1 (ii
9 1 3 P (x = 1) = f (1) = 1 4 4 1
8
= 0. 225
3 1 " " (iii P (x ≥ 2) = p (x = 2) + p (x = 3) + … + p (x = 9 ) = 1- [ p (x =0) + p (x = 1)] = 1 – 0. 07 5 – 0. 225 = 0. 7
148
! A
7 8 3 9 8 " x D (8
( x / 4 E 5 3 3 % 4 5
5 ... 2 E 0، 1 n= 5
8
p=q = 12
9 & 8 H 6 8 % I 9 . 8
; 5 4 3 2 1
(0 0) (1 1) (2 2) (3 3) (4 4) (5 5) ; ) 7 H ( 9 8 1 5 5 x 2
(0 , 0) = P(0) p(0)
()
0 5 5 1 1 = 0 2 2
()
5 1 = 50 2
L. "
()
1 0 1 5 5 0 2 2
2
8 % I 9 . 8 ∴
5 2 1 P = 50 + 2
()
2
()
()
()
1 5 1 5 5 5 5 + + 1 2 3 2 2 2
149
2
1 2
5 2
()
()
1 5 1 5 + 54 + 55 2 2 2
2
1 1 1 1 1 1 + 25 + 100 + 100 + 25 + 210 210 210 210 210 210
= =
252 210
=
252 63 = 1024 256
x / 4 E 3 25 3 G " x D (9 q =
4 5
E
25 ... E 1 E 0 ! ( p=
1 5
E
n = 25
4G 9 ( % 9 8
" ( % 9 8
P = (x = 4) + p (x = 5 ) + … + p (x = 25 ) = 1 – [ p (x = 0) + p (x = 1) + p (x = 2) + p (x =3)]
( )
( ) 15 45 + (252 ) 15
25 1 4 025 + 125 5 5 =1- 3 22 + 25 1 4 3 5 5
( )
24
2
4 5
23
= 1- (0. 0037 7 7 7 + 0. 0236110 + 0. 07 08 336 + 0. 135 7 644) = 0. 7 66
3 K Z ) I
Z ) 2 9 2 G ( % G ( 0) G 9 K ( 8 150
( 6
(10
( 8
( 6 I
Z ) 9 K ( 0
?
()
1 0 2 4 p = 04 3 3
:; )
()
1 1 2 0 11 3 3
2 1 16 16 = = = 3 3 35 243 4
P=
2 2 2 2 1 16 × × × × = 3 3 3 3 3 243
E 1 E 0 :! ( x / 4 " x D =
6 ...
p = 2 2 K
= q = 2 M K 6= n
2 4G 9 ( % " 6 5 4 3 2
P (x ≥ 2) = 1- p (x < 2)
= 1 – [ p (x = 0) + p(x = 1)]
H 2
151
(11
0. 30 0. 70
()
()
P = 1- [ 60 (0.3)0 (0.7)6 + 16 (0.3)1 (0.7)5 ]
= =
1 –( 0 .1 1 7 6 4 9 + 0 .5 7 9 8 2 5
0 .3 0 2 5 2 6 )
9 2 2 2 2 + + 2 H 6 * 2
/ 4 3 + + 1 " x D 2 4G
(12
! ( x
(0 E 1 E 2 E ... E 10)
0. 1 = p = D 4 =
0. 9 = q = D 4 / = 10 = n
= D 4 / A P = 1 –
4 = D 4
% & A
(100 ) ( 0 . 1 )
0
( 0 . 9 )10 = =
1 – ( 0 . 9 )10
4
0 .6 5 1 3
np ; ) ! 0 " * + * 4 ! ) )
npq ; ) ! 0 " * +
(13
n p = 9 … … … … … … … . (1 ) n p q =
18 … 5
… … … … …
(2 )
( 2) ( 1)
152
9 q =
18 5
q =
18 2 = 45 5
p =1-
2 3 = 5 5
n= 9 ÷
3 45 = = 15 5 3
(1 ) ! P ( 4 $
%
n = 35
(14
p = 0. 2
q = 0. 8
np = * 4
npq = µ = np ∴ µ = 35 (0. 2) = 7
σ 2 = npq
σ 2 = 7 (0. 8 ) = 5 . 6
* + x 2 E F ( 8 ! 5 G " x D (15 ( 8
! J ) )
λ =
232 =1 232
153
(i
e − λ λx x!
f ( x) =
p ( x= 2 ) =
e −1 (1) 2 2!
f(2 ) =
= e-1/ 2 =
0 .1 8 3 9
e = 0 .3 6 8 -1
P ( x<2 ) =
1 e −1 (1) x
∑
= =
(ii
x!
x =0
=
C 2
e-1 ( 1 +
1 ) 11
2 e-1
0 .7 3 6
) * + & % & ! C " x D (16 4 = 3 % ; H
e − 4 ( 4) 0
f ( 0 )= = P (x ≤ 3) =
(i
=
e
-4
0!
0 .0 1 8
∑
(ii
e - λ (λ) x x!
154
= p (x = 0) + p (x = 1) + p (x = 2) + p (x = 3) f(0) = e f(1) =
-4
= 0. 018
e−4 4 1!
= 0. 07 2
f(2) =
e − 4 (4) 2 = 0. 144 2!
f(3) =
e − 4 (4)3 = 0. 19 2 3!
f ( x ≤ 3) = 0. 426
% ! 1 ! x D (17 λ =
2 = ( 4 ! % ) 5
2 = , 0 4 5 ! % )
4 = , 0 4 10 ! % ) , 0 4 I !
i) P (x = 0) = f(0) =
e − 2 20 0!
=e
-2
= 0. 135
ii) λ = 4
, 0 4 !
P (x > 4) = 1-P (x ≤ 4)
155
λ = 2∴
λ = 4
e - λ λx P (x ≤ 4) = ∑ x! x =0 4
= P (x = 0) + P(x =1) + P(x =2) + P (x =3) + P(x =4) f(0) = e
-4
f(1) = e
-4
f(2) =
e − 4 (4) 2 =8 e 2!
f(3) = e f(4) =
(4)
-4
-4
64 6
e − 4 (4) 4 64 = e 4! 6
-4
=
0. 018
=
0. 07 2
=
0. 144
=
0. 19 2
=
0. 19 2
∴ P (x > 4) = 1 – P ( x ≤ 4 )
0. 618
= 1 – 0. 618 = 0. 38 2
4 . 8 ! ( & 1 " x D (18 λ = 2 3 % ) * + &
e − 2 24 P ( x = 4 ) = f (4) = 4
=
2 e 3
=
2(0.135) 270 = 3 3
-2
= 0. 09 0 156
( 8 ! ( 5 G 1 " x D (19 λ = 4 3 % ) * + x 2
i)
e − 2 24 P (x = 0) = f (0) = 0!
= e ii)
= 0. 018
-4
P(x ≥ 2) = 1 – P (x < 2) = 1 – [ P (X = 0) + (P(X =1)] = 1 – (0. 018 + 4 e
-4
)
= 1 – (0. 018 + 0. 07 2) = 0. 9 1
/ 4 10 × 6 3 ! K % " x D (20
6 =
(6) (10) = λ 3 % ; H ) * + x 2 10
f(x) = i)
e − λ λx x!
P (x = 0) = f(0) = e
0. 0025
157
-6
=
ii)
P (x ≥ 1) = 1- P (x < 1)
= 1-P (x = 0)
= 1 – f (0) = 1 - 0. 0025 = 0. 9 9 7 5
500 2 ! [ " x D (21 2 .5 =
500 = λ ∴ 200
f (x) =
i)
P (x≤2 ) =
f(0 ) +
f(0 ) = e
-2 . 5
f(1 ) = e
-2 . 5
f(2 ) = P ( x ≤2 ) =
ii)
f(1 ) +
f(2 )
e − 2.5 (2.5) 2 = 3.126 e - 2.5 2
6 .6 2 5 e
-2 . 5
=
( 6 .6 2 5 )( 0 .0 8 2 ) = 0 .5 4 3
1 – P ( x < 3)
P ( x ≥ 3) =
1 - {f (0) + f(1) + f(2)}
= =
1 –0 .5 4 3 =
e - λ λx x!
0 .4 5 7
158
/ 7 × 3 ( 4 ! K % " x D (22
f (x) =
e - λ λx x!
11.5 =
i)
P ( x= 0 )= f ( 0 )= e
ii)
P ( x ≤1 ) = f ( 0 ) + f ( 1 ) f(1 ) = e
-1 1 . 5
f ( 0 )+ f ( 1 )= 1 2 .5 e
-1 1 . 5
21 =λ 2
= 0 .0 0 0 0 1
1 1 .5 -1 1 . 5
159
= 0 .0 0 0 1 2 6 6
16 0
The Normal Distribution
2 " ( 2 % + / ! % * + %
2 ! 2 7 8 * ' 3 E , . 9 % )
H 2 J "( % " % + " 3 ) , 4 E 5 8
% + ) 3 % + / = % ? @ H E 3 4 .
/ 2 3 / , ) 5 % + ( +
2 2 " ! 2 * + H / ) E 3 4 . H J (7 ! 3 % ) ) 3 7 ( 8 K . M 3 % % + F % $
D e M o i v er ; / % * + H
H 2 % 1809 / ! G au s s I M / % 7 % 1733 /
H 2 & G au s s D i s t r i bu t i o n I 2 M * 2 + ; 3 ) L ' * + ( 2 8
( 2 2 2 ( ' 3 8 * + . ! 0 " * + 2 " T % +
2 / E ! % 9 9 * + H 9
2 1 2 ( 2 " ( 2 9 8 J ; = * + 9
" ! % * + 9 E * + H 3 2 ; H ! 0 %
2 . 2 2 " K ) ; ) ; H ! ) ) ! )
9 6 7 7 3 4 3 ! ) 4 . ) 16 1
2 & ! 2 G 7 K ) ) ( &
- 8 ; ) 9 ( ) @ H * (3 % 2 1 ; 2 ( 2 " 3 9 ( ) ! : 9 " 1
. 8 ! 0
1
2 2 & ( % ! 0 & @ µ (* 2 4 ) ! 2 ) 2 ) 2 2 ( 4 K)
$
% & ' %
2 ! 2 ( 2 % , 4 E σ ; %
4 E 2 ! ! ) ) ! & ; % 3 ! ; % ! ) ) ,
16 2
µ= 2
µ= 4
x
µ= 6
2
f(x) σ = 0 .5 σ =1 σ=2
x
µ= 3
3
9 ; µ ! ) ) " ! % 9 (1 .3 9 9 " ! ) ) ) 9
9 6 9 3 ) / ) 8 ! ! ) ) (2 . ) ) 4
:( A
. ; ) 9 3 ) D (3
! % 9 ? σ µ ( 4 & (4
µ + σ µ R σ * 3 ) % 68. 26 !
16 3
R
µ R2 σ * 3 ) % 95. 46 ! RK µ+ 2σ
3 σ µ R 3 σ * ( ) % 99. 74 ! RQ µ+
1. 96 σ µ R1. 96 σ * ( ) % 95 ! R µ+
2. 58 σ µ R2. 58 σ * ( ) % 99 ! R2 µ+
:! 4 ! - '
%68.26
13.6%
34 .13%
34 .13%
13.6%
2.14%
2.1 4 %
x
668.26 8 .26
95.46% 9 9 .7 4%
3 H 6 % D + 3 x 8 ! 0 % 1
:( ( 1 8 7 % 3 ( "
16 4
f (x) =
1
σ 2π
e
−
1 x - µ 2 2 σ
C
−∞ > x < ∞
! % 1 ! ) ) ! µ
2. 71828 3 4 " e
σ2 3 2
3. 14159 ; ) ( ( ) ! π
µ ! 2 ) 3 ) ; H ! % * + K
. " 2 N( µ , σ2 ) + K ( σ ; ) ; % 3 )
20 ; ) (* 4 ) ! ) ) 3 % 1 ! % N (2 0,4 ) .(2 ; ) % ; ) 4 ; )
2 ( 2 ) D 2 ( " 3 f (x)
. 8 0 L 1 - 8
) 3 7 H 9
< x < b ) b , a " 4 * σ2 3 µ 3 ) x=
f (x) 3 9 7 8 ( ) ; ) p (a
b,x= a
f(x) µ
16 5
a
b
5
The Standard Normal
D i s tri b u ti on
1 9 ! ) ; % ! % 1 / ) , 1 9 ; 3 8
! ) 3 ) ; H ! %
! 3 ( " 3 z + 3 + N(0.1) 1 e- z 2π
f (x) =
2 /2
−∞ < z < ∞
f (z) 9 ( ) D ( " 3 f (z)
∫
∞ -∞
1; )
1 -z 2 / 2 e d z= 1 2π
z z ; % ! % ! 0 % 1 D 4
f 2 ( " 3 9 7 8 ( ) z = b E = a
; ) ; z = b E z = a (z)
P (a < z < b ) =
b
∫
a
1 2π
2 e− z / 2 d z
! ; % ! % 9 ( ) ( ) ) ., 2 ! 7 %
4 8 ! % 1 D 4 K )
f (x) 2 ( 2 " 3 9 7 8 ( ) K) ) % ' f(x) 16 6
X 2
7 M 4 H J & σ2 µ ! )
4 L ' H J 3 ! 8 1 ! 0 % 1 6 3 % 4 ) K ) M 3 ? T ; 7 M
* 2 ) x , σ2 , µ 3 / ! % 9
9 2 6 N ( µ , σ2 ) ! 2 % 1 ( % 8 7 H 9 K 1 ! 7 3 ) K ) N(0.1) ; % 1
., 2 ! ) 7 %
N ( µ , σ2 )
N( 0 . 1 )
? N ( µ , σ2 ) ! % * + 3 x ! 0 % 1 H 6
; 2 % ! 2 % * 2 + 3 2 z= 3 8
x−µ σ
! 0 % 1
! ) 3 ) % % L % + 3 ; N(0.1)
9 6 3 σ2 3 µ ! ) 3 ) ! % 1 ; 1 ! 2 % * + H & z=
x−µ 3 ; % ! % σ
1
7 7 8 ( ) K ) ; % ! % * + 9 6 .6 !
P (x ≤ b) =
x / 4 ( % ( 4 b H 6
P (x - µ ≤ b - µ ) 167
x−µ b−µ P ≤
= =
σ
b−µ P z ≤
σ
σ
f(x)
µ
168
b
x
f(z)
Q
z =
6
z
b−µ σ
z N ( µ , σ 2 )
P (X ≤ b) = P (Z ≤
! 9 3 = M ( )
b-µ ) σ
P (X > b) = 1 – P (X ≤ b)
'
b-µ ) σ
= 1 – P (Z ≤
(7 ! ) a< b H 6
P (a ≤ X ≤ b) = P (Z ≤ a ) 9 6 ( ) A
169
b-µ a -µ )- P (Z ≤ ) σ σ
4 b ) 9 6 ( ) ; ) ;
x
a b
µ
7
8 ( " 3 9 " ( 8
i)
x
∫
−∞
f(z)d z = =
ii)
0
∫
−∞
1
2
f(z) d z +
∫ f(z) d z
x o
∫ f(z) d z
x
+
o
−x
∞
−∞
x x
∫ f(z) d z = ∫ f(z) d z =
1
2
-
∫ f(z) d z
o
2 ) / ( ) ! ! @ H
3 2 x / ( K ) ; % ! % 9 .( ) x / Q ) &
170
: :1
2 N (3 E 4 ) * + 3 ! % ! 0 1 X H 6
O5 E 3 X *
:
x−µ σ 3−3 z1 = =0 2 5−3 z2 = =1 2
z=
; % * + 9 6 * +
p (3 < x < 5) = p (0 < z < 1)
! ! !
z= 1 z= 0 3 ) ; )
0
1
0. 3413 ; ) , 2 !
171
:2
N ( µ,σ2 ) ! % * + 3
X ! 0 % 1 H 6
i)
P ( µ − σ < x < µ + σ)
ii)
P ( µ − 2σ < x < µ + 2σ )
iii) P ( µ − 3σ < x < µ + 3σ) i)
z1 = z2 =
:
:
(µ − σ ) − µ = −1 σ (µ + σ) − µ =1 σ
∴ P ( µ − σ < x < µ + σ) = P( −1 < z < 1)
z=1
z=1
z = -1 ( ) F
z=0 ( ) + z=0
-1
0
1
µ=0 ! ) ) ) 9
z=1
z=1 ( ) =
Z 9 " (
z = 0 ( ) ; ) z = 0 z = -1 ( )
0. 3413 ; ) 7 . ! 172
∴ P ( µ − σ < x < µ + σ ) = 0. 34 13 +
0. 34 13
= 0. 6 8 26
'
= 6 8 . 26 %
* 9 ( ) D % 68. 26 ! % H
.! ) ) ; %
ii)
P ( µ − 2σ < x < µ + 2σ ) (µ − 2σ) − µ = −2 σ (µ + 2 σ ) − µ z2 = =2 σ
z1 =
∴ P ( µ − 2σ < x < µ + 2σ) = P( −2 < z < 2)
F 8 ( ) + z = 0 z = -2 7 8 ( ) ; )
z =2 z =0
= 0. 4 7 7 3 + 0. 4 7 7 3 = 0. 9 5 4 6
'
=9 5 .4 6 %
* 9 ( ) D % 95. 46 ; . 8
! ) ) %
Z 173
-2
iii)
2
(µ − 3σ) − µ = −3 σ (µ − 3σ) − µ =3 z1 = σ
z1 =
∴ P ( µ − 3σ < x < µ + 3σ) = P(−3 < z < 3)
F 8 ( ) + z = 0 z = -3 7 8 ( ) ; )
: z = 3 z =0
= 0. 4 9 8 7 + 0. 4 9 8 7 = 0. 9 9 7 4 =9 9 .7 4 %
3 2 '
* 2 9 2 2 ( ) D % 99. 74 ; . ! ) ) 3 %
99.74%
-3 3
174
:( ; % ! % * + .1
(0. 2580)
P (0 < z < 0. 7)
(0. 3849)
P (0 < z < 1. 2)
(0. 8643)
(ii
P (R1 < z < 0)
(iii
P (R0. 82 < z < 0. 96)
(iv
(0. 3413)
(0. 2742)
(i
P(
P(
(0. 6254)
z < 0. 6R)
z > R1. 1)
(v
(vi
z = R0. 5 ( ) .2
: N (2, 1) ! % * + 3 x ! 0 % 1 H 6 .3 (0. 0227)
P (x > 4 )
(0. 4773)
P (0 < x < 2 )
(i
(ii
2 & 2 % 3 2 + % T K. H 6 .4
3 2 3 8 68. 50 ! ) 3 ) ; H ! % * + :3 8 2. 3 ; %
175
+ ( % K ;
(i
/ & H ( % ! 3 ( ) !
(ii
.(0. 0643) (3 8 72) / 4 6
(0. 1935) 3 8 72 70
2 ! % * + H K 3000 H 6 .5
2 / 2 ) 170 ; 2 ) 2 G 7 H & ! ) ) : / ) 5 ; ) & ; %
./ ) 185 " / & H ( ( ) ./ ) 185 / & + H (
2 ) , 2 / & H ( ( ) H 6 2 0. 2881 x % 4
(i
(ii
(iii
.O H
3 2 ) ; H 2 ! 2 % * + 3 x ! 0 % 1 H 6 .6 : J 0. 30 ; % 3 80 ! ) P (X ≤ 80. 36)
P ( X ≤ C) = 0. 95 H 6 C % 1
4
& 4 $
(i
(ii
G W * 8 .7
* + & * 8 H Q G 4 7 / )
E / 2 0. 0025 ; % 3 / 2. 5 ! ) 3 ) ! % 176
2 2 4 2 ! 2 a ( 0 ( ) !
O/ 2. 5049 2. 4951
; % ( ) ! 4 % ) H 6 .8 ( 2 ) ! % % + * G H 6 0. 50 O( ) ! 3 2. 5 ' H %
2 ( ) 1500 ! 0 & 8 ) H 6 .9 : % % + * ( ) 50 ; % .( ) 1400 4 , ) 8
(i
.( ) 1550 " S % ) 8 6
(ii
.( ) 1550 1450 S % ) 8 6 (iii I 2 10 ; ) 3 3 + , 8
.10
3 + + I 1. 5 ; % (O U NCE S )
12. 4 7. 9 2 & + * ! ( ) ! %
OI
* 2 + 2 & ( ) ! T 6 H 6
5 ; 2 % 3 2 100 3 ) ; H ! % : .
. 100 + ) (
. 100 ) ( 177
.11 (i
(ii
110 100 2 ) ( (iii .
178
( )
.7 , 2 ( .1 ! 2 2 ( 2 2 ( 2 2 ) .2
2 ! 2 ( 2
= ( )
R0. 5 R ∞ 2 F 8 0.5 2 ( 2 ) " ! 2 ( 2 ) A
4 ∞ 9
= 0. 5 9 8
0. 5 – 0. 1915 = 0. 3085
; % * + 9 6 ! % * + .3 z= z=
z -µ σ
4-2 =2 1
∴ P (X > 4 ) = P (z > 2)
0
= 0. 5 – 0. 4 7 7 3
2
Z Z1 =
179
= 0. 0227 -0 . 5
0 - 2 = -2 1
(ii
(i
∴
Z2 =
P (0 < X < 2)
2 - 2 =0 1
= P (z > 2) = P (-2 < z < 0) = P (0 < z <2)
=
0. 4 7 7 3
.4 x -µ σ 72 - 68.5 3.5 = = 2.3 2.5
.i
Z =
= 1. 5 2
∴ p (x > 7 2) = p (z > 1. 5 2)
0
= 0. 5 -0. 4 35 7 = 06 4 3 70 - 68.5 = 0. 6 5 2.3 72 - 68.5 z2 = = 1. 5 2 2.3
.ii
z1 =
∴ P (7 0 < x < 7 2) = P (0. 6 5 <
z < 1. 5 2) 18 0
= 0. 4 35 7 – 0. 24 22 = 0. 19 35
185 − 170 =3 5
z=
. i .5
∴ P (x > 18 5 ) = P ( z > 3 )
= 0. 5 – 0. 4 9 8 7 = 0. 0013
3
0
3000 = %
100
x = %
0. 13
(0.0013) (3000) 0.100
x=
= 3. 9 ≅ 4 z =
0 . 28 8 1
.ii
x
- 170 5
7 = z ( 4
.iii
x
0.8 ! 0. 2881 ( )
∴ 0. 8 =
0
x = 18 1
χ - 170 5
174
Z =
.i .6
80 - 36 - 80 = 1.2 0 .3
∴ p ( x ≤ 80.36 ) = p ( z ≤ 1.2)
= 0. 5 +
1 .2
0. 38 49
= 0. 8 8 49
0 .4 5
z=
.ii
c − 80 0 .3
z ( 2 4 2
0
– 0. 50) ( 2 2 ) ( 2 2 0
! 0. 45 ( ) ; (0. 95
144424443
1. 64
0.95 =
0.4 5
∞ − ∞ ! " ∞
∴
c − 80 = 1.64 0 .3
∴ c = 8 0. 49 2
z1 = z2 =
2.4951 − 2.5 = 1.96 0.0025 2.5049 − 2.5 = 1.96 0.0025
.7
∴ P (2. 49 51 < x < 2. 5049 ) = P
(-1. 9 6 < z < 1. 9 6 )
-1. 9 6 1. 9 6
= 2 (0. 4750)
182
= 0. 9 5 z1 = z2 =
-3 2
.8
2 .5 − 4 =-3 0 .5 3−4 =-2 0 .5
∴ p (2. 5 < x <3) = p (-3 < z <-
-
2)
= p (2 < z < 3) = 0. 49 8 7 – 0. 4773 = 0. 0214
z=
1400 - 1500 = -2 50
.i .9
∴ p (x < 1400) = p (z < -2 )
-2
= 0. 5 – 0. 4773
0
= 0. 0227 z=
1550 - 1500 = 1 50
.ii
∴ p ( x > 1550) = p (z > 1 )
0
= 0. 5 – 0. 3413
1 183
= 0. 158 7
-1 1
z1 =
1450 - 1500 = -1 50
z2 =
1550 - 1500 50
.iii
=1
∴ p (1450 < x < 1550) = p (-1
< z <1 ) 0
= 2 (0. 3413) = 0. 6 8 26 7.9 - 10 = 1.4 1.5 12.4 - 10 z2 = = 1.6 1.5
.10
z1 =
-1. 4
0
∴ p (7. 9 < x < 12. 4 ) = p (-1. 4
< z < 1. 6 )
1. 6
= 0. 419 2 + 0. 4452 = 0. 8 6 44
.11 .i
184
z=
100 - 100 =0 5
p ( x > 100 ) = p (z > 0 ) = 0. 5
z=
.ii
100 - 100 =0 5
p ( x < 100 ) = p (z > 0 ) = 0. 5
.iii
100 - 100 =0 5 100 - 100 =2 z2 = 5
z1 =
∴ p (100< x < 110) = p (0 < z <
2)
= 0. 4773
185
2 ) 2 3 & ( 0 ! ( 0 % ( R1 2 K ) .F ) ) I K ) G 5 Z
. ( 0 % (3 /8 )
= 0. 5 2
P(A2) = 0. 7
S ! 2 % c ! " A2 E A1 H 6 R 2
P(A1)
: P(A1 A2) = 0. 3
" A2 A1
(0. 9)
( ; ) & D ) P ( A 1 A 2)
(0. 2)
P ( A 2 R A 1)
(i
(ii
(iii
P ( A1 R A2) (iv
( ) )
= 0. 4 2 S ! 2 % c ! " P(B) = 0. 7
(0. 8)
B
A H 6 R3 P(A)
: K ) P(AB) = 0. 3
B A " D 4
186
(i
& I B A D 4
(ii
AD 4 /
(iii
(0. 5)
(0. 6)
(0. 1)
BD 4 / AD 4
(iv
: ( C 7 7 ! R4
(6/36)
7 = 4 D A1
(i
10 " = 4 D A2 (ii (3/36)
5 ; 2 ) 2 2 4 2 = 4 D A3 (iii (10/36)
4 G 9 7 7 & = 2 % A4 (iv (11/36)
(3 /6 )
: ( K ) , ) I ! R5
6 G $
! + D $
9 ! + D
9 6 ! "
(i
(ii
(3/18)
(4/6) 5 4 D H 6 ! + D (iii (1 /6 )
4 G $
9 6 ! "
(9/18) ; G H 6 ; D
187
(iv (v
( 8 H + M 9 6 % & 4 8
. + 2 1 ( 8 6 K ) J
3 R6
1 ! + 1 ( 8 6 ! & 3
(5 /9 )
2 % 50 & % % 20 P ) A H 6 R7 2 % 90 2 ! 4 B P ) 5 &
2 & % 2 K K ) H ? .5 &
.5 K K )
(0. 82)
5 2 3 K ) K H 6 , ) Z ) ! R8 O 3 ! ! A
(10/18=0. 5555)
/ 0. 40 = 5 8 F ! K H 6 R9 (0. 2592)
(0. 92224)
(0. 98976)
(0. 07776)
:
K . 5 K -
(i
4G 9 K -
(ii
-
(iv
" G 9 K . 4 - (iii
! C . " 5 D 9 8 R10 O 6 ! +
(0. 17342)
188
2 0. 3 & [ 8 ( 8 6 H 6 R11
9 & ( 8 6 ! & 4. 6 / + . P 8 .% 80 4 G
0. 7 = J
(0. 7)n = d P 8 1-(0. 7)n > 0. 8 K
(0. 7)n > R0. 2 (0. 7)n < 0. 2
(0. 7)1 = 0. 7 , (0. 7)2 = 0. 49 (0. 7)3 = 0. 743 , (0. 7)4 = 0. 2401 (0. 7)5 = 0. 16 8 07
∴ n = 5 P 8
189
:
19 0
(1)
e-x x
e-x
x
e-x
x
0.0
1 .000
2 .0
0.1 35
4.0
0.1
0.9 05
2 .1
0.1 2 2
0.2
0.8 1 9
2 .2
0.3
0.7 41
0.4
e-x
x
e-x
x
0.01 8
6.0
0.002 5
8 .0
0.00034
4.1
0.01 7
6.1
0.002 2
8 .1
0.00030
0.1 1 1
4.2
0.01 5
6.2
0.002 0
8 .2
0.0002 8
2 .3
0.1 00
4.3
0.01 4
6.3
0.001 8
8 .3
0.0002 5
0.67 0
2 .4
0.09 1
4.4
0.01 2
6.4
0.001 7
8 .4
0.0002 3
0.5
0.607
2 .5
0.08 2
4.5
0.01 1
6.5
0.001 5
8 .5
0.0002 0
0.6
0.549
2 .6
0.07 4
4.6
0.01 0
6.6
0.001 4
8 .6
0.0001 8
0.7
0.49 7
2 .7
0.067
4.7
0.009
6.7
0.001 2
8 .7
0.0001 7
0.8
0.449
2 .8
0.061
4.8
0.008
6.8
0.001 1
8 .8
0.0001 5
0.9
0.407
2 .9
0.055
4.9
0.007
6.9
0.001 0
8 .9
0.0001 4
1 .0
0.368
3.0
0.050
5.0
0.0067
7 .0
0.0009
9 .0
0.0001 2
1 .1
0.333
3.1
0.045
5.1
0.0061
7 .1
0.0008
9 .1
0.0001 1
1 .2
0.301
3.2
0.041
5.2
0.0055
7 .2
0.0007
9 .2
0.0001 0
1 .3
0.2 7 3
3.3
0.037
5.3
0.0050
7 .3
0.0007
9 .3
0.00009
1 .4
0.2 47
3.4
0.033
5.4
0.0045
7 .4
0.0006
9 .4
0.00008
1 .5
0.2 2 3
3.5
0.030
5.5
0.0041
7 .5
0.00055
9 .5
0.00008
1 .6
0.2 02
3.6
0.02 7
5.6
0.0037
7 .6
0.00050
9 .6
0.00007
1 .7
0.1 8 3
3.7
0.02 5
5.7
0.0033
7 .7
0.00045
9 .7
0.00006
1 .8
0.1 65
3.8
0.02 2
5.8
0.0030
7 .8
0.00041
9 .8
0.00006
1 .9
0.1 50
3.9
0.02 0
5.9
0.002 7
7 .9
0.00037
9 .9
0.00005
19 1
e-x
! % 9 ) /
(2 ) / 2 4 , 2
( 2,0 A)
0 Z 0.00 .01 .02 .03 .04
A 0.0000 .004 0 .008 0 .01 20 .01 6 0
Z 0.4 7 .4 8 .4 9 .5 0 .5 1
A 0.1 8 08 .1 8 4 4 .1 8 7 9 .1 9 1 5 .1 9 5 0
Z 0.9 4 .9 5 .9 6 .9 7 .9 8
A 0.326 4 .328 9 .331 5 .334 0 .336 5
.05 .06 .07 .08 .09
.01 9 9 .0239 .027 9 .031 9 .035 9
.5 .5 .5 .5 .5
2 3
.1 9 8 5 .201 9 .205 4 .208 8 .21 23
.9 9 1 .00 1 .01 1 .02 1 .03
.1 .1 .1 .1 .1
0
.039 8 .04 38 .04 7 8 .05 1 7 .05 5 7
.5 .5 .5 .6 .6
7
.21 5 7 .21 9 0 .2224 .225 8 .229 1
.1 .1 .1 .1 .1
5
9 6 36 5 4
.6 .6 .6 .6 .6
2 3
9
.05 .06 .06 .07 .07
.20 .21 .22 .23 .24
.07 .08 .08 .09 .09
9 3 32 7 1 1 0 4 8
.6 .6 .6 .7 .7
7
.25 .26 .27 .28 .29
.09 8 7 .1 026 .1 06 4 .1 1 03 .1 1 4 1
.7 .7 .7 .7 .7
2 3
30 .31 .32 .33
.1 .1 .1 .1
.7 .7 .7 .8
1
2 3 4 6 7 8
7 5
1 7 21 25 29
1 4
5
7
9 3
4 5 6 8 9
0 1
4 5 6 8 9
0 1
4 5
9
6 8
7 0
z
Z 1 .4 1 1 .4 2 1 .4 3 1 .4 4 1 .4 5
A 0.4 207 .4 222 .4 236 .4 25 1 .4 26 5
.338 9 .34 1 3 .34 38 .34 6 1 .34 8 5
1 .4 1 .4 1 .4 1 .4 1 .5
6
.4 .4 .4 .4 .4
27 9 29 2 306 31 9 332
1 .04 1 .05 1 .06 1 .07 1 .08
.35 .35 .35 .35 .35
08 31
1 .5 1 .5 1 .5 1 .5 1 .5
1
.4 .4 .4 .4 .4
34 35 37 38 39
.2324 .235 7 .238 9 .24 22 .24 5 4
1 .09 1 .1 0 1 .1 1 1 .1 2 1 .1 3
.36 .36 .36 .36 .37
21
1 .5 1 .5 1 .5 1 .5 1 .6
6
.4 .4 .4 .4 .4
4 06 4 1 8 4 30 4 4 1 4 5 2
.24 .25 .25 .25 .26
8 6
.37 .37 .37 .37 .38
29 7 0 9 0 1 0
1 .6 1 .6 1 .6 1 .6 1 .6
1
8 0 1 2
1 .1 1 .1 1 .1 1 .1 1 .1
.4 .4 .4 .4 .4
4 6 3 4 7 4 4 8 5 4 9 5 5 05
.26 .26 .27 .27 .27
4 2 7 3 04 34 6 4
1 .1 9 1 .20 1 .21 1 .22 1 .23
.38 .38 .38 .38 .39
30 4 9 6 9 8 8 07
1 .6 1 .6 1 .6 1 .6 1 .7
6
.4 .4 .4 .4 .4
5 1 5 5 25 5 35 5 4 5 5 5 4
.27 28 .28 .28
9 4 23 5 2 8 1
1 .24 1 .25 1 .26 1 .27
.39 .39 .39 .39
25
1 .7 1 .7 1 .7 1 .7
1
1 8 4 9
19 2
4 5 6 7 8
5 4 7 7 9 9 4 3 6 5 8 6 08 4 9
4 4 6 2 8 0
7 8 9
0 2 3 6 5 7 8 9
0 2 3 4 5 7 8 9
0 2 3 4
5 7
0 2 4
.4 5 6 4 .4 5 7 3 .4 8 2 .4 5 9 1
.34
.1 331
.8 1
.29 1 0
1 .28
.39 9 7
1 .7 5
.4 5 9 9
.35 .36 .37 .38 .39
.1 .1 .1 .1 .1
36 8 4 06 4 4 3 4 8 0 5 1 7
.8 .8 .8 .8 .8
2 3 4 5 6
.29 39 .29 6 7 .29 9 6 .3023 .305 1
1 .29 1 .30 1 .31 1 .32 1 .1 33
.4 .4 .4 .4 .4
01 5 032 04 9 06 6 08 2
1 .7 1 .7 1 .7 1 .7 1 .8
4 .4 .4 .4 .4
6 08 6 1 6 6 25 6 33 6 4 1
.4 .4 .4 .4 .4 .4 .4
.1 .1 .1 .1 .1 .1 .1
5 5 4 5 9 1 6 28 6 6 4 7 00 7 36 7 7 2
.8 .8 .8 .9 .9 .9 .9
7
2 3
.307 9 .31 06 .31 33 .31 5 9 .31 8 6 .321 2 .3238
1 .34 1 .35 1 .36 1 .37 1 .38 1 .39 1 .4 0
.4 .4 .4 .4 .4 .4 .4
09 9 1 1 5 1 31 1 4 7 1 6 2 1 7 7 1 9 2
1 .8 1 .8 1 .8 1 .8 1 .8 1 .8 1 .8
.4 .4 .4 .4 .4 .4 .4
6 4 9 6 5 6 6 6 4 6 7 1 6 7 8 6 8 6 6 9 3
0 1
2 3 4 5 6
8 9
0 1
Z 1 .8 8 1 .8 9 1 .9 0
A 0.4 7 00 .4 7 06 .4 7 1 3
Z 2.4 1 2.4 2 2.4 3
A 0.4 9 20 .4 9 22 .4 9 25
Z 2.9 4 2.9 5 2.9 6
A 0.4 9 8 4 .4 9 8 4 .4 9 8 5
1 .9 1 .9 1 .9 1 .9 1 .9
.4 .4 .4 .4 .4
7 1 9 7 26 7 32 7 38 7 4 4
2.4 2.4 2.4 2.4 2.4
.4 .4 .4 .4 .4
9 27 9 29 9 31 9 32 9 34
2.9 7 2.9 8 2.9 9 3.00 3.1
.4 .4 .4 .4 .4
1 .9 6 1 .9 7 1 .9 8 1 .9 9 2.00
.4 .4 .4 .4 .4
7 5 0 7 5 6 7 6 2 7 6 7 7 7 3
2.4 2.5 2.5 2.5 2.5
9
.4 .4 .4 .4 .4
9 36 9 38 9 4 0 9 4 1 9 4 3
3.2 3.3 3.4 3.5 3.6
2.01 2.02 2.03 2.04 2.05
.4 .4 .4 .4 .4
7 7 8 7 8 3 7 8 8 7 9 3 7 9 8
2.5 2.5 2.5 2.5 2.5
4
.4 .4 .4 .4 .4
9 4 5 9 4 6 9 4 8 9 4 9 9 5 1
2.06 2.07 2.08 2.09 2.1 0
.4 .4 .4 .4 .4
8 03 8 08 8 1 2 8 1 7 8 21
2.5 2.6 2.6 2.6 2.6
9
.4 .4 .4 .4 .4
2.1 2.1 2.1 2.1
.4 .4 .4 .4
8 26 8 30 8 34 8 38
2.6 2.6 2.6 2.6
4
.4 .4 .4 .4
1 4
2 3
1
5
4
2 3
4 5 6 7 8 0 1
2 3 5 6 7 8
1
0
7
6
5
2 3
6 7 8 9
0 1
2 3 4 5 6 7
... Z 3.4 7 3.4 8 .4 9
A 0.4 9 9 7 .4 9 9 8 .4 9 9 8
9 8 5 9 8 6 9 8 6 9 8 7 9 8 7
3.5 3.5 3.5 3.5 3.5
0
.4 .4 .4 .4 .4
9 9 8 9 9 8 9 9 8 9 9 8 9 9 8
.4 .4 .4 .4 .4
9 8 7 9 8 8 9 8 8 9 8 9 9 8 9
3.5 3.5 3.5 3.5 3.5
5
.4 .4 .4 .4 .4
9 9 8 9 9 8 9 9 8 9 9 8 9 9 8
3.7 3.8 3.9 3.1 0 3.1 1
.4 .4 .4 .4 .4
9 8 9 9 9 0 9 9 0 9 9 0 9 9 1
3.6 3.6 3.6 3.6 3.6
0
.4 .4 .4 .4 .4
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
9 5 2 9 5 3 9 5 5 9 5 6 9 5 7
3.1 3.1 3.1 3.1 3.1
.4 .4 .4 .4 .4
9 9 1 9 9 1 9 9 2 9 9 2 9 9 2
3.6 3.6 3.6 3.6 3.6
5
.4 .4 .4 .4 .4
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
9 5 9 9 6 0 9 6 1 9 6 2
3.1 7 3.1 8 3.1 9 3.20
.4 .4 .4 .4
9 9 2 9 9 3 9 9 3 9 9 3
3.7 3.7 3.7 3.7
0
.4 .4 .4 .4
9 9 9 9 9 9 9 9 9 9 9 9
19 3
2 3 4 5 6
1
2 3 4 6 7 8 9 1
2 3 4 6 7 8
1
9
2 3
2.1 5
.4 8 4 2
2.6 8
.4 9 6 3
3.21
.4 9 9 3
3.7 4
.4 9 9 9
2.1 6 2.1 7 2.1 8 2.1 9 2.20
.4 .4 .4 .4 .4
8 4 6 8 5 0 8 5 4 8 5 7 8 6 1
2.6 2.7 2.7 2.7 2.7
9
.4 .4 .4 .4 .4
9 6 4 9 6 5 9 6 6 9 6 7 9 6 8
3.22 3.23 3.24 3.25 3.26
.4 .4 .4 .4 .4
9 9 4 9 9 4 9 9 4 9 9 4 9 9 4
3.7 3.7 3.7 3.7 3.7
5
.4 .4 .4 .4 .4
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
2.21 2.22 2.23 2.24 2.25
.4 .4 .4 .4 .4
8 6 5 8 6 8 8 7 1 8 7 5 8 7 8
2.7 2.7 2.7 2.7 2.7
4
.4 .4 .4 .4 .4
9 6 9 9 7 0 9 7 1 9 7 2 9 7 3
2.27 3.28 3.29 3.30 3.31
.4 .4 .4 .4 .4
9 9 5 9 9 5 9 9 5 9 9 5 9 9 5
3.8 3.8 3.8 3.8 3.8
0
.4 .4 .4 .4 .4
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
2.26 2.27 2.28 2.29 2.30
.4 .4 .4 .4 .4
8 8 1 8 8 4 8 8 7 8 9 0 8 9 3
2.7 2.8 2.8 2.8 2.8
9
.4 .4 .4 .4 .4
9 7 4 9 7 4 9 7 5 9 7 6 9 7 7
3.32 3.33 3.34 3.35 3.36
.4 .4 .4 .4 .4
9 9 6 9 9 6 9 9 6 9 9 6 9 9 6
3.8 3.8 3.8 3.8 3.8
5
.4 .4 .5 .5 .5
9 9 9 9 9 9 000 000 000
2.31 2.32 2.33 2.34 2.35
.4 .4 .4 .4 .4
8 9 6 8 9 8 9 01 9 04 9 06
2.8 2.8 2.8 2.8 2.8
.4 9 7 7 .4 9 7 8 .4 9 7 9 .4 9 8 0 .4 8 0
3.37 3.38 3.39 3.4 0 3.4 1
.4 .4 .4 .4 .4
9 9 6 9 9 6 9 9 7 9 9 7 9 9 7
2.36 2.37 2.38 2.39 2.4 0
.4 .4 .4 .4 .4
9 09 9 1 1 9 1 3 9 1 6 9 1 8
2.8 2.9 2.9 2.9 2.9
.4 .4 .4 .4 .4
3.4 3.4 3.4 3.4 3.4
.4 .4 .4 .4 .4
9 9 7 9 9 7 9 9 7 9 9 7 9 9 7
0 1
2 3 5 6 7 8 0 1
2 3 4 5 6 7 8 9 1
0 2 3
9 8 1 9 8 1 9 8 2 9 8 3 9 8 3
19 4
6
5
4
2 3
6 7 8
1
9
9
8
7
6
4
2 3
19 5
2 ) 2 3 & ( 0 ! ( 0 % ( R1 2 K ) .F ) ) I K ) G 5 Z
. ( 0 % (3 /8 )
= 0. 5 2
P(A2) = 0. 7
S ! 2 % c ! " A2 E A1 H 6 R 2
P(A1)
: P(A1 A2) = 0. 3
" A2 A1
(0. 9)
( ; ) & D ) P ( A 1 A 2)
(0. 2)
P ( A 2 R A 1)
(i
(ii
(iii
P ( A1 R A2) (iv
( ) )
= 0. 4 2 S ! 2 % c ! " P(B) = 0. 7
(0. 8)
B
A H 6 R3 P(A)
: K ) P(AB) = 0. 3
B A " D 4
19 6
(i
& I B A D 4
(ii
AD 4 /
(iii
(0. 5)
(0. 6)
(0. 1)
BD 4 / AD 4
(iv
: ( C 7 7 ! R4
(6/36)
7 = 4 D A1
(i
10 " = 4 D A2 (ii (3/36)
5 ; 2 ) 2 2 4 2 = 4 D A3 (iii (10/36)
4 G 9 7 7 & = 2 % A4 (iv (11/36)
(3 /6 )
: ( K ) , ) I ! R5
6 G $
! + D $
9 ! + D
9 6 ! "
(i
(ii
(3/18)
(4/6) 5 4 D H 6 ! + D (iii (1 /6 )
4 G $
9 6 ! "
(9/18) ; G H 6 ; D
19 7
(iv (v
( 8 H + M 9 6 % & 4 8
. + 2 1 ( 8 6 K ) J
3 R6
1 ! + 1 ( 8 6 ! & 3
(5 /9 )
2 % 50 & % % 20 P ) A H 6 R7 2 % 90 2 ! 4 B P ) 5 & 2 & % 2 K K ) H ? .5 &
.5 K K ) (0. 82)
5 2 3 K ) K H 6 , ) Z ) ! R8 O 3 ! ! A
(10/18=0. 5555)
/ 0. 40 = 5 8 F ! K H 6 R9 (0. 2592)
(0. 92224)
(0. 98976)
(0. 07776)
:
K . 5 K -
(i
4G 9 K -
(ii
-
(iv
" G 9 K . 4 - (iii
! C . " 5 D 9 8 R10 (0. 17342)
O 6 ! +
19 8
2 0. 3 & [ 8 ( 8 6 H 6 R11
9 2 & ( 8 6 ! & 4. 6 / + . P 8
.% 80 4 G
0. 7 = J
(0. 7)n = d P 8 1-(0. 7)n > 0. 8 K
(0. 7)n > R0. 2 (0. 7)n < 0. 2
(0. 7)1 = 0. 7 , (0. 7)2 = 0. 49 (0. 7)3 = 0. 743 , (0. 7)4 = 0. 2401 (0. 7)5 = 0. 16 8 07
∴ n = 5 P 8
19 9
:
! " # $ % &
. ! ' ( ' ) ' (
+ ( " + , - . , / 0 . 1
2 3
4 , / 5 ' 6 7 " 1 8 9 1 : 2 5 , ) , 6 7 ' , ( ; ' 0 . '
+ ' < ( 8 < (
= , > 1 = ( , 7 ( ? -
8 9 1 # 1 . > @ 6 &
% 5 A #
+ 7 .( A , - # ) > 1 8 , + 6+
! < D ' . D - < (
) - (
. ( ,
.
.
1
2
PROBABILITIES
"
. " (E v e n t )
! " # ! $ % & ' % ( ) !
, & % % + ! (* ) % '
" - & ' $ . / 0 1 % $ 2 % $ $ . /
4 % % 5 , ! .+ " ' 3 & $ 1/4 . $ - (52) .
4# $ "& " $ -
$ $ - % , 9 % % . $ & ' : % % ; '
" 1/4 & ' & $ - < # = % > - % % % ? ' " 4#
.13 .
$ & '
"& "
@ . 1/ 4 =
5 , 4
13/52 . ' , 9
B % , ( . 2 > 2 0 % 2
, 9 1 5 # & ' 5 , $ / " % ! 3 3 " = - & ' 2 % " % ! % & 4# 4
& ' $ # $ C % ' D , . " % ! % ' & ' $ = E
& ' '
0 < 9 2 - " C & ' $ = E
+ # < 9 ? # ' E 4! & ' = ' 4 < # . $ #
& % (F E R M AT ) " ' (PASCAL ) ! ) " C
3
$
" & ' + $ % " % ! % . . (BE R N O U L L I ) . # & '
( % # # " / % 0 F
:
! % & !
: , . & ' E & F
: 1
! / * % (* ) % . & ! $ + ; % % % F
'
5 4 3 2 1 $ : @ / & . $ % ! H I %
K 5 3 1 J J ' - / % % F
1 % 6
< K K K K. & K $ K # ; K ' , K ! . . $ K
! " < % . . J ' - / (E x p e r i m e n t ) < K / $ % ! " > $ (E v e n t ) .(S a m p l e S p a c e )
. & % L 1 ) ! $ ! - 3 / * + M
=
(P o s s i b l e C a s e s ) # 2
$ % / ! & $ 1 # E H I % " & .
% ! $ # $ - & % (* ) ' $ % $
5 4 3 2 1 % ! % . & % $ !
6 $ # $ - & $ & ' 2 $ % ! "
' 6 . % . & $ & '
(F a v o r a b l e C a s e s ) # 3 4
. J , 3 < 9 J N & " H I % & .
& ' J ' - < # = . 3 ! , ; ' % . > C & .
3 , .
& " ; ' % .
& $
" < $ ) * ) " D , . 5 3 1 < # =
.$
(E q u a l l y L i k e l y C a s e s ) $ " # 4
& ' $ % $ % = $ % " ! %
! , 9
% ! % 0 ! & ' . % C 0 1 % $ ' ) ! ! ! J $ # ) " ! " ! D , . ; ' ( # E
.4 & ' 4 = % 0 1 % %
(M u t u a l l y E x c l u s i v e E v e n t s ) % 5
. ) , 9 ' % % B A )
" - & ' < # = ! % & % (* ) '
.
(I n d e p e n d e n t E v e n t s ) $ & 6
. 9 B - ! , 9 # ) B A )
$ # $ - & % (* ) ' . E O B - & ' ) N : -
.< @ $ % ) ? $ % ) $ $ % ; ' (E x h a u s t i v e E v e n t s ) $ ' 7
! , 9 $ & ' $ # 2 3 ... C B A 3 <
.$ + 9 % . 9 3 5
! , 9 : $ ' $ 4 E % (* ) '
1 # : % @ $ # 2 3 " D , . E % E < # = ; ' 5 , ! ." 1 = D , . $ 1 =
: !
% & % 6 5 4 3 2 1
.. 9 3 : % @ $ # 2 3
" ( ) ( & * + & " < # $ 3 $ $ < # 1 =
$ 3
: D
: ! .B - ! @ 3
. J / % P
. & P
(T h e o r e t i c a l A p p r o a c h )
$ ) 3 J / % & ! * ! (C
F
' < # 3 $ % ! "
< 9 $ "
$ % . .3 & ' + & ' ! 4 = % " ! : 1, "
3 8
! " # $
11 = $ % ! "
3 = $ "
6
: ,
3/11 = + C ! 4 ∴
$ ' $ ) 3 4 , 9
. $ "
3 & ' + & ' ! 4 = % & $ % ! " :2 , "
. " K! 2 Q K9 9 0 # + C (4) 5 % . K! , 9
& ' $ ! 2 ) % % ) E 2 (D C B A)
: $ " N
A C E . .
D A C E . .4 D A C E . .R
A C E . .
J $ % ! " $ . S & % L 1 : , S = {AB, AC, AD, BC, BD, CD}
" 3 . A C E $ "
P ." 6 $ % ! " $ "
= $ % ! "
P(A ) =
3 1 = 6 2
D A E $ " 5 5 % . P(A or D) = P(AUD) =
7
5 6
P4
D , A E $ $ 5 % . PR P (A , D) = P (A ∩ D) =
1 6
A E $ " 3 5 % . P
∴ P (A ) = A E & .
3 1 = 6 2
P( A ) 3
* J / % & ' " = 4 F
% 9 4# 0
5 % .
# + ( (
$ % : # " , E SP %
! (* ) '
. K 4 +
& ' 0
* &
*
4% % # R 4 +
& ' 0
$ % B - 4
*
"
$ % B - 4
$ "
) & & . K 4 +
% B
$ % ! "
) & % B 4%
& % D , .
4 ( 4= : % Q % % % ! .* J / % 4= 8
:%
1
# , " - . + 2
, . ! $ % ! " ) : 2 J / % '
$ : $ ' S
E 2 E % (* ) ' I
% % ! * ' # R 4 & . (: )
4 ! 1/4 J T* ) (* : % !
U C : % @ J / % E , .
.$ # ) $ % ! " D , .
1/2 J ( ! , ! ? > % 5 , !
" ( ! , : % ! & . $ $ " < #
+ % > & ' . 2 : % @ 5 , (< ) % ! , ) $ % !
.3 % M < # ! ,
.
D , . ) < # 4# V * E W < 9 $ " ? 2 % % . .& . , . " =
(E m p i r i c a l A p p r o a c h )
! N " * J ! 2 < # " ! $ # % , 9 & "
n1 < # < 9 ! : / : /
% & "
% & "
n2 : /
%
; ' n3 V =
n1 = ! : # "
$ % N 9
n2 = : # "
$ % N n3 = V = : # "
$ % N 1 $ % ! " ) K # E 4% D , . 3
$ - % 4% D , . " - # ! ! N " % ! # !
.& < # % 10 % 20 % 70 (* ) ! - $ ) ! 3
: < # = .
L im
" ! $ # & % ! @
N
: < # = .
L im
" ! $ # & % @
N
: < # = .
L im
" ! $ # & % V = @
N
: /
n1 = 0.70 N → ∞
n2 = 0.20 N → ∞ n3 = 0.10 N → ∞
"
! n $ # $ - % , 9 5 , ! r 1/2 # E - $ % D , . . = =
$ % ; ' r . = n r $ % " - # ! n " # ! : % " ) ! V = n " % ! , 9 n
1/2 ( $ - $ % U = ( ! n ! % : % 3 1/2 L im N 10
n1 = 1/ 2 N → ∞
J
. % $ - & % =
< # = .
:
N
' r ! " # $ % & ( ) ! * + $ m
r
Li
. K 3 < % $ ' % 3 .... A3 , A2 , A1 3 " % ! , 9
J N K
D , . 3 ; ' & Q E @ 3 J 3 $ < 9
; ' ( ! ( 3
$ # ! ! ( * +
! , * + * ( ) (1) ! 2 & ' !
' % ) A B ! , ; '
(1)
s A
B
P (A B) = P(A) + P(B) ; '
) B - (C
P (AU B) = P (A) + P (B)
11
YJ '
< # = . % . & $ & ' : 3
3 5 3 1 < # = D % J '
< # = : ; ' $ ' % $ ) * ) 3 D , .
P (1 3 5 ) = P (1 ) + P (3 ) + P (5 )
1 1 1 + + 6 6 6
= =
1 2 :4
Y 2 < # = . % . & %
:
(2 2) (1 1) < # = D % 2 < # = 1 % ! $ ' % 3 & . (6 6) ... (3 3) 36 P ( 2 ) = P (1 1)+ P (2 2)+ ... + P (6 6)
1 1 1 + + ... + 36 36 36 6 = 36 =
12
% B, A P (A) + P(B) = 1
B, A P (A) = P (B) = 1/ 2
A A A # $ A ! " #
% &
% ' %
A) 3 = ! B A ) & ' % 2 %
B A ) B - 1 % < # B B - 1 % < # A B - (B
& (2) ! 2 U C ! " - & ' (2)
s
A and B
B
A
(' C A 3 # $ " B ) P (A) + P (B) O
K K! $ K/ * 4K ! B 3 # $ " B 9
$ K " C B 3 # $ 5 # A 3 # $ " > K % K% % ; ' P (B) P (A) > $ & ' : % ; ' , ( B A B - 13
= % P (A B ) [ , P (A B) : . , . P (A B) < #
P (A B )= P (A) + P (B) – P (A B)
P (AU B) = P (A) + P(B) – P (A ∩ B)
: 5
. " ! 2 Q 9 9 0 # + C ) < 9 B
Q 9 9 0 # + C 4 D C A C E Y " N & ' # ) (D C B A . ) " ! 2 :
$ % ! " B . & % L 1
{AB,
S =
AC, AD, BC, BD, CD}
(AB, AC, AD) & . A E $ " P
(AD, BD, CD) & . D E $ " P (AD) & . ( D A E $ " P
.( # % # = & $ % < # = % " > - 4 P (AU D) = P (A D) = P (A) + P (D) – P (A ∩ D) = 3 3 1 + − 6 6 6
14
:6
! "
< # = ) B
< # = ) A F
1 %
:
"J %
" ! $ % $ $ -
40 52 13 52
P (A) = P (B) =
P (A ∩ B) = P (AU B) =
10 52
" > % -
P (A) + P (B) – P (A ∩ B) 40 13 10 + − 52 52 52 43 52
= =
( ) * + , - *
+ . , $ / . , $ $ 0 1 2 .
; ' B A # ) %
! , 9
P (A ∩ B) = P (A) P (B) 15
: 7
Y % R & % (3 3) : < # = .
:
1 . % @ & Q 3 : < # = 6
1 . % & % ) & Q 3 : < # = 5 , ! 6 P (A ∩ B) =
P (A) P (B) = 1 1 1 × = 6 6 36
Conditional Probability
$ M "
, E " ! 2 +
- B E W () ! , 9 A 3 3 $ ' $
& I * 9 2 % - , E
( # ! (* ) ' 3 (= E 2 . $ ' ! : % ; ' 1 # & ' $ #
1 # & ' * M . 2 = - ! , 9 (A 3 ) $ # J 2 .(B 3 )
< B 3 B - 2 A 3 B - ; ' , ! .
A 3 P (A/B) =
16
< # 4! & 2
.B 3 B -
:& ) & 2 1 U C % : 8
+ C " ! 7 + " ! 3 < # J 0 !
4 < 9 % , ; ' . 9 D < # ! ! : % % % % F
1 %
$ ! " ; ' B + ! 4 < 9 A + C !
:& # ) # = % &
AA
C !
P
+ C $ % ) + < @
P
+ C $ % ) + C < @
AB BA
P
!
BB
P
C 9 ! 4 $ % ) ) < @ ! 4 H I % "
P(A/A) = 6/9
3 4 5 P(A) = 7/10
P(B) = 3/10
A
:& 5
A P(B/A) = 3/9 P(A/B) = 7/9
B
A
B P(B/B) = 2/9 17
B
=
:& # ! : ! 4! ; ' , ! .
+ C $ % ) + C < @ ! !
! 2 + C $ % ) ! & ' C + C < @ !
.+ C < @
P (A ∩ A) =
P (A , A) = P (A) P (A/ A) 7 6 42 × = = 10 9 90
P (A ∩ B) = P (A) P (B/ A) =
7 3 21 . = 10 9 90
P (B ∩ A) = P (B) P (A/ B) =
3 7 21 . = 10 9 90
P (B ∩ B) = P (B) P (B/ B) =
3 2 6 . = 10 9 90
K)
:& 2 * & % ) , . % O
. /
-
1 $ 2 3 ( ) P (B) B , A 0 #
. B ! * + . / A ! . /
P (A/B) = P
(A ∩ B) P(B)
:
( ) B ! * + . / A ! . / ( B ! B , A
18
4 ) + 3
P(A B) 4! H % % % % ! D * & 2 P (A ∩ B) = P (A) P (B/ A) = P (B) P (A/ B) ! 3 3 $ & 1 ' ) ) ! @ $ V = D , . % % % ! P (A ∩ B ∩ C) = P (A) P (B/ A) P (C/ AB) : 9
3 : % " + " ! 3 + C " ! 7 < # J 0 !
Y+ C # ! ! . 9 " !
< @ 4 & ' + C ! < # = .
& % ) 4 & ' + C ! < # = .
3 ) 4 & ' + C ! < # = . P (A1 ∩ A2 ∩ A3) =
:
A1
!
A2 A3
% # 2
P (A1) P (A2/ A1) P (A3/ A1A2)
=
=
7 6 5 10 9 8 210 720
19
: 1 0
. 9 4# $ - "
Y 2 !
< @ $ - & ' 10
< # = . $ % ) $ - & ' 10
< # = .
:
A
!
B
P (AB) = P (A) P(B/A) =
4 3 . 52 51
! 4 : " # 5 / 3 &
' . 7 # 6 # 6
J " 4 " # D , . ) = $ %
:& ) & ' $ C $ % 2 @ $ 3 : 1 1
:& # ! R $ ) * ) %
. % 2 %
' $ 2 6 I R
.$ % 3 %
' $ 2 8 II R
.& % 1 %
' $ 2 4 III R
20
$ I 2 $ ( 2 : % 4 $ I 2 $ E
.( 2 ! (P) . ' .(C :
$ $ I 2 $ # ) , . & ' . R $ ) * ) E %
. (B) ( (A) ( % ! 2 E %
( i)
(ii)
B ' ! & $ # D , . $ $ % 2 )
: 2 #
1 /3 1 /3
2 /6 4 /6
I
3 /8
II
1 /3
5 /8 1 /4
III 4 C
3 /4
A B
A B
A B
= ( $ " J 3 !
( 2 E ) II R E (* ) ' . , . & ' B ' ! " 1 3 . . ( % 3 8
B ! & 2 < 9 J N $ ' % " $ ) * ) : % 3
= J ) 4 # . $ ) * ) " D , . "
:(& 2 < #
21
P
1 4 1 5 1 3 . + . + . 3 6 3 8 3 4 =
1 4 5 3 + + 3 6 8 4
= =
1 49 49 = 3 24 72
(Baye s ' T h e ore m )
D 1 .$ $ # " 1 # E - % =
%
% C '
. % 4%
- % = ( I 2 " : . @ % = " - $ 1 D , . !
& $ / % - E % B % , . $ # I < # $ ] < # + % F
1 $ =
.& 2 * ( . !
" 4 < 9 $ / %
" # @ Q ! 4V & ' .$ $ % " # $ = E 2 : E * E 9 $ % $ 3 $ ' C 9
E < # & " .$ ? 3 , . Prio r
& C * E
& # < $ H I % < # = - $ = E 2
< # $ " " *
< # " (* ) ' Pro b a b il it y
% .& # * $ # ) # R% M $ ! -
J < $ ! $ % " # + C
< # 4
. Po s t e rio r Pro b a b il it y
22
% % . , & ' ! - > I 2 @ $ % (* ) ' :& . F
> I 2 @ !
1
^
A2
> I 2 @ !
^
^^
^^
^^
^^ ^^
^
^^
^^
^^ ^^
^^
^
^^
^ ^
10
^^
^^
A3
^^
^^ ^^
^^
'
A1
> I 2 @ !
2 3
1 D , . , $ = E F
^^
^^
^^ ^^
A10
.$ ' % $ # 2 $ ) " D , .
4% % . / D , . B $ % " % "
, 9
E ! - > .
< , . ; ' P(A4/B) F
% % % 9
' % # 2 ) ) ! @
.'
' % # 2 ) < # $ / %
S # : 2 $ $ # /
$ P (D)
≠ O
8
9 # -
0 B A
! : 2 ; 2 # $ ! ( D
P (A/ D) =
P ( A D) P(A D) + P(B D)
P (B/ D) =
P ( B D) P(A D) + P(B D)
S
D B A 23
A
A B % # C 1 $ %
& ' % E 0 . 40
: 12
! , 9
, 0. 30 # ) + % ! , 9 B B % # C 1 % -
$ I 2 $ % E , ; ' .B # C 1 , 0 . 40 A # C 1
YA B % # C 1 ! . ' " % ! % E :
:& . " F
A B % C 1 E
B B % C 1 E
C 1 . A B % 2 (*
C 1 . B B % 2 (*
E ! E !
C 1 . A B % 2 D E !
C 1 . B B % 2 D E !
1 %
P (A) P (B) P (M/A) P (M/B) P (F/A) P (F/B)
:& ! 2 C $ % ! " " M
P(M /A) = 7 /10
P(A) = 4/10
P(B ) = 6 /10
A
P (A F ) = P (A ) P (F /A ) = F
P(F /A) = 3 /10
P (A M ) = P (A ) P (M /A ) =
P(M /B ) = 6 /10
M
P(F /B ) = 4/10
F
4 3 . 10 10
=
28 100 12 100
=
P (B M ) = P (B ) P (M /B ) = 6 /1 0 . 6 /1 0 = 3 6 /1 0 0
B
24
4 7 . 10 10
P (B F ) = P (B ) P (F /B ) =
6 4 . 10 10 =
24 100
- E
P (A/F) =
P (AF) 12/100 = P(AF) + P (BF) 12/100 + 24/100 =
6
6
12 36 24 36
12 100 24 100
) , $
1 .0
12 1 = 36 3
:& " D , . ! ) % % % !
2
6
6
3 10 4 10
4 10 6 10
7
36 100
$-
1 .0
A B
8
Repeated Trials
H I% < # = " 4
" (% (
$% $
$ 1 ! , . & ' 0 %
! < # 4 ! & $1 # E
.$1 # E / " / 0 1 % "
9 $ 6 9 - * 5 2 8 :+6
% % ? ' =
: /
% % ! N % $ - + 9 % (* )
( $!
=
< # = % ' =
: / "
$1 # E H I% 5 % . ? / * %
5 % . J ( $! N =
N P1) ( $!
1 = =
1 = ) ... ( $! = 25
N)
NP2)
" ; ' " 0 E % $ - % (* ) ' . $% ! $ N+ 1 $! 1 =
$!
!
$! 3
$! 4
$! 5
=
=
5
=
4
=
2
=
=
:& . $% !
3
1=
1
" " $% ! " B J
" , $# ! 4 & ' " D , . ) 4
.
& I% ) & % E% " )
B in o m il P r o b a b ilit L a w
:& O J
% % , . < 9 = $ 1 ! U C
½ = $ - < # = /
. " 3* ) $ - $ ( > - 3 < # = / /
½ . ½ . ½ = (½)3
! " 0 E $ - % > - 0 E % , ; '
< # . / : 1 % " - & ' & % % > - 3* ) < # =
> - 3* ) < # = /
! 5 , .(½)2 = J -
= " - 0 1 % & ' ( - < # . / (½)3 (½)2 26
< @ "
3* ) & ' = /
$ % & .
D,.
!
J & ' = / 4 % % % ! - & ' . / > 4 / % F
V - J & ' . / > - 3* )
/ J ! 4 # U C
.
> - 3* ) < # . / ! > - 3* ) J < # =
"
4 % ' E% 4 # < # = #
> - 3* ) $ ! ! 3 > - 0 E % ! ! & ! 3 > - 0 E % ! ! & '
4 % J '
( =
! D * D % J , ⊂ 53 J > - 3* ) $ ' !
& ' ! " ) J 3* ) ' D , . $ : ' ! < # = % ' 4 & ' '
4 C % 5 , $ !
⊂ 53 (½)3 (½)2
3 B - % $% $
= 1 0 (½)5 = 10 32
$ & ' : % % ' % > % O
(1-P) : - (P) 3 , . B - ! : -
3 (r ) & ' 3 , . B - ; ' (n) $
⊂ nr p
r
D , . % !
= ! ( n> r >
(1 -p )n-r
0)
& ' E ' % ! , !
& I% ) & % , . - = B - ' " ) " , $# " 1
27
2
= $ ! & ' " )
" 0 E % $ - & . % ) < 9 :( 13)
/ $= E " ; ' > - 0 E &
:& # ! % , . E ! :
= " 5 = /
⊂5 (1 2 )
(1 2 )0 = (1 2 )5 = 321
⊂ 4 (1 2 )
(1 2 )1 = 5 (1 2 )5 = 325
5
5
4
5
= " 4 = /
= " 3 = /
⊂3 ( 1 2 )
(1 2 )2 = 10 (1 2 )5 = 10 32
⊂ 2 (1 2 )
(1 2 )3 = 10 (1 2 )5 = 10 32
3
5
= = /
2
5
5 1
⊂
= = /
(1 2 )1 (1 2 )4 = 5 (1 2 )5 = 325
⊂0 (1 2 ) 5
0
= = /
(1 2 )5 = 1 (1 2 )5 = 321
28
& I% ) 5 ! 1 & ' $ & . H I% D , . / * %
(12 + 1 2)5
- n .
& ! 4 & ' 3 B - "
! 8 ; ' $ ' % $# 2 " & . n ... 2 1 1 =
, E?
P(O) + P(1) + P(2) + ... + P(n ) = 1
: 14
& % 4 3 $ / $% ! " =
." 3 %
$% ! " :
3 : / P D / P D / P
" 3* ) D / P
= ( 3 : / P
⊂
31 0
125 5 = 216 6 6 0
3
1 2 31 5 1
⊂
= D / P
1 25 75 = 3 = 6 6 6 36 216
29
⊂
31 2
15 5 = 6 6 216
⊂
31 3
2
1
= D / P
= " 3* ) D / P
1 5 = 216 6 6 3
0
$ ' % $# 2 " D , . @ 5 , = " B / * %
& I% ) 5 ! 1 & ' $ & . H I% D , . / * % !
1 5 + 6 6
3
- * 5 2 8 : +
" ! 4 $ & ' " % + & '
+ , . & ' " ) " , $# " & ' J / 0 1 %
B - ' V & ! 4 $ & ' " 0 % % E% " D , .
) & ' Q E < 9 $
3
. J
Hypregeometric Low
:& ) % , . U C % % % !
: 15
: & . # D > $# ) ! N : # E 0 ! % , E R .
" ! % + ! n3 + 3 0
! n2 + C ! n1
$ " ! ! 4 4 # 9 .+ ! r3 + ! r2 + C ! r1
"
.
N
⊂R
:
.
0 ! ! R E
, . n1 K + C " ! + C ! r1 4 3 $% ! n2 K + " ! +
! r2 4
n3 + " ! + ! r3 4
⊂r
2
n2 r2
n3
⊂r
(⊂ )(⊂ )(⊂ )
.
.
, . 1
n2
.
3 J n1 r1
n1
⊂r 3
.
, .
n3 r3
$ "
. , .
= ! &
n1
n2
n3
1
2
3
⊂r ⊂r ⊂r N ⊂R
3K K
R = r1 + r2 + r3 = n1 + n2 + n3
:
@ # ! n1 % $# ) ! N : # E 0 ! %
! , 9
R 9 % Z # ! nz ... & % ) # ! n2
3 1
# ! r1 $ " ! ! ; ' !
: . z # rz ... & % ) # r2 @ n1
n2
nz
1
2
z
⊂r ⊂r ... ⊂r N ⊂R
J & % < $V = D , . :16
% 20 $ I 2 : % " , E " 5 4 * 10 ' $-
: . '
.$ !
(i
." < # J (ii
:
= $ ! J % (i 10
< # $
5
⊂2 ⊂1 = 225 15 455 ⊂3
< # J & % "
4 * ( % ! S
10
< # J (ii
-% J , . -@ 5
⊂0 = 335 = 1 − ⊂3 15 455 ⊂3
" 3* ) $ < # J . 5
10
⊂1 ⊂2
+
5
10
5
10
⊂2 ⊂1 + ⊂3 ⊂0 15 ⊂3 3 2
=
335 455
Expectation
" $ / %
V & ' S 4 C
% F
V $# .
& C > -
E2 [ % < 9 P % C ' , ; ' $ I
= ; ' : % % : # = J , a # $ - < 9 X . V D , . : # S
E2 > - < P.X
4 ! × . = > -
E > - # %
& .
∴
E = P.X
i " % ! n > ' χ a # ! , 9 , .
= > - # $ $ ; ' I1
P.X (1 + i) − n =
P.X
(1 + i) n
: 17
$- 4 , 9 % % 5 4 * U . 4 $ & '
B % $ $- " % ! ( $- 52) 4 # $ Y 4 $ # ! + : ' 4 1
4
J , ) . ' "& " :
= "& " $- 4 = U
. > - $ - < # = % % ! I $ , . % C
, ; '
: ! )
E = P.X.
= 1 4 (5) = 1.25
. %
> %
J : ! J 3 3
:18
4 : ' / % ( . J ) & C > - a # !
Y 0. 4 0 4 ! ! , 9 %
100 D - . 4 ! & ' 4
40 × 100 = 40 = & C > - 100
E = P.X. =
5 > ' ) & ' ! , . $ - F
:
:19
1
= & C > - # $ $ ; ' % 3 . 5 I1 ! " %
E = =
= =
P.X
(1 + i) n 40
(1.035)5
40 1.1876862 3 3 .6 7 8 9 = 3 3 .7
< 9 . $ - > ' " & $ " , 9 : % , . < %
%
3 3 . 7 % ! - * $ % " ! 2 = I1
$! $ % I' > - ) " ! 2 D , .
$#
; ' " % 0 E
5 I' ! %
100 & . . $ - a # > ' $ ' ! ! ( % 3 . 5
." %
3 4
: V J , a # & . & C > - + 2 $ - " % !
& C > - ! ' ; ' . $ - . > - a # 4 ! $ ! & '
$ - > ' ? " ! 2 , 9 < # ? ! ' 0 D , .
$ > ' , ( % ? ) * ( : # N a # ) . .
? F
V & C > - # $
$! I1 0 < # & C > - # $ $ & . :2 0
$! 2 2 & ' S
E2 .
& ( % % 3 . 5
Y @ a # , 9 %
2000 a # : > ' ? " ?
. , E ! - < # + $ -
" # $ ! @ E " & & ' "
& $ 9 8 7
& . O <
( .S
:
& .
186 7 !
E2 100000 .
$ / % $ ' $ = - < # + l P = x+n lx
x % - < # -
& . 1x
x + n % - < # -
& . 1x + n
3
7 - & ! @
l 78106 P = 40 = = 0.84314 l 20 92637
3 5
& . & C > - # $ $
P.X
=
0.84314 × 2000 (1.035)
20
(1 + i) n
=
1686.28 = 8 4 7 .4 6 7 1.9897877
< # % N 2 & ' S
E2 ! < # , . < %
%
84 7. 4 6 7 : - % - ! 2 ? $! 2 < 9 > '
@ 5 , . @ a # , 9 %
2000 a # : > ' ? . 20 ( % % 3 . 5 -@ " ! 2 D , . . ) $! 2
( / ! %
2000 a # > ' & 1 !
! : % ; ' $% . @ < ! 2
4 0 < # " ) E & ' > - -
) a [ 5 , ) $ - E # $
.H % # > - 4 $ - E # $ B
> 2 & ' %
1000 a # ) & ' S
3 & .
E2
: 2 1
% D - ( ) ( & $ % =
% !
& '
E
U " :& # !
½ . J < # = P1
< @ $%
2 . J < # = P1 5
$ % ) $%
½ . %
( 1050) D - < # = P2
3 6
%
( 1102.5) D - < # = P2 3 . 5
' % 5 . $ @ & ' : I1 % # , ; ' % ( % # : I
Y ) +
9 ) +
S
E2 , . U = %
9 - , E :
:& # ! % $# ! 2 ) 0
1/ 2
2/ 5 3/ 5
10 0 0 1/ 2
1 05 0
2/ 5 3/ 5
0 1102. 5
0 1102. 5
4 : # = > - E # $ 4 % 5 ,
:& # ! 5 , ) a # $ - % [ % ) " # E 1. 0 – 10 0 0 2.
1102.5
(1.05) 2
=
− 1000 =
1050 − 1000 = 1.05 1102.5 1050 4 . + − 1000 = (1.05) 2 1.05
3 .
3 7
- 10 0 0 0 0 10 0 0
:& # ! 5 , " 4 % 5 ,
@ =
1 2 2 . = 2 5 10
& % ) =
1 3 3 . = 2 5 10
3 ) =
1 2 2 . = 2 5 10
> =
1 5 3 . = 2 3 10
" $ - E # E $ 4 % -
:& # ! 5 , > - E J > - 4 % % ! : % ' $#
1/ 2
0
3/ 5
10 0 0 1/ 2
2/ 5
1 05 0
2/ 5 3/ 5
0
1000 0
1102.5
2/10
200
3/10
0
0
0
2/10
0
1102.5
1000
3/10
300 100
$
% 100 : : # = > - E # > - $ - . ) +
3 8
; S
E2 , . U = % % % % ? ' , 9
3 ($- 52) 4# $ - " .1 1 4
:$ O " 4 .$ %) 4 - < @ $-
@ # - !
.
1 (J % ) ! 2 - ! .4 16 1 4
! 2 0 1 % - ! .R
( ' R ! % E . SE 2 @ R $) * ) % .2 1 4
3 8
: . ' (
0 % E $) * ) ' @ !
.
# % ! .4
4 $%! H I % = " 0 E % $ - " .3 > - $) * ) $ & ' =
$) * ) /
3 9
. % !
. .4 1 Y % 8
> C =
< # = , ; ' % $ - & SE 2
> C : %? ' $! < # = , 9 %= ) n $ # D, . SE 2 ! , ; '
.5
& ' + " ! 3 * )
.+ C ! + !
+ C ! D, . ! 4' . %= ! 4 1 6
! + " ! 3 * ) % $#) " ! 5 : @ ! .6
" ! > + " ! 3 * ) % " ! 7 : & %) C
% ! . ' 0 ! ! ! " .+ C (26/35)
Y + C -@ < #
+ " ! 3 + " ! 6 + C " ! 3 < # J 0 ! .7 . ' 9 ( I 2 " ! 3 " (1/10 )
Y ! %
4 + " ! 3 + C " ! 5 % $#) ! 12 : 0 ! .8 :$ O " 4 0 ! " ! 3 " + " !
(21/55)
(27/55)
+ " ! /
.
+ ! / .4
(34/55)
-@ < # + ! ! .R
(3/11)
) ! ! 5 %. 0 : % .D
(3/44)
#! " ! ! .
40
$- # % $- 10 0
! ' I 1 $- # % 50 . - I
"* 1 & ' > .9 "= = E
Y $-
! 30 + C ! 45 % # $#) ! 10 0 : 0 ! .10
0 ! < 9 " %
! " + ! 25 +
! " ) 0 ! < 9 " %
$ %) ! " )
< # $ 3 * ) " ! ! . $) ) (0 . 0 3375)
Y F
4
" ) ($- 52) 4# $ $- " .11 ( 1 / 2)
(1/52)
(10 /169)
:$ " 4 .$ %) $- "
< @ $- 0 1 % $ %) $- !
.
< @ $- 0 1 % & . $ %) $- ! .4
0 1 % - ! .R
(1/13) = 0 1 %
0 1 % - ! .
(3/169)
= 0 1 % - ! .D
$ " 3 % $ - " .12
+ ) * ) " & ' = & ' !
=
3 . ( 5 / 8)
3 . ( 1 / 4)
41
.
.4
3 * ) "
< 9 4. , , 9
2 : # 0 E ? ! , 9 .13 3
. ' . : 4. , , 9 1/6 $#' 4. , , 9 1/4 ( 2 : # E , 9 @ [ =
& ' : # E ? (13/36) Y $ I 2 $
% 8 $ % 5 < # J @ - %=
% ! , 9 .14
. ' .$ # 11 $ % 6 < # J & %) $ # Y ( - %= ( I 2 4 % ! (0 . 36878)
4 # 1 $ : #I E & . $ ! .15
4 ) $ D, & % L 1 4! . # 0 % ( 1 / 4)
(5/16)
(15/16)
:$ "
"% $#I %
% ) ! $#I %
) ! @ < # "% 3 $#I %
.
.4 .R
! , ; ' B , A = -% < 9 + % "! 2 Q 9 " .16
$= -% < # = 0 . 6 . A $= -% < # =
. ' 0 . 1 . ( = -% < # = 0 . 3 . B (0 . 8)
:
B $= -% A $= -% < # $! 2 = P 42
P ( AB) =
1 5
3 S & % L 1 & ' ) B , A B .17 P (B/A) =
3 2 , 5 5
1 2
P (A/B) =
1 3
P (B) , P (A) KK
M2 M1 "%! "%! , 9 $ I ! U = > %= & ' .18 R%M B 0 . 40 0 . 30 0 . 30 4 < # > %= M3
4 < # 3 * ) "% ! R%9 $I 2 3 1 ! , 9
: % @ R%9 ( I 2 = 4 .4 R%9 .
Y M1 $% ! > %=
[ = , . ! . ' 4
. (8/20 9/20 3/20 ) Y M3 > %=
Y M2 > %=
9 ($- 52) 4# $ "- 5 % , 9 .19 $- < # $E @ D, . + . ' .(3243/10 829) Y 4 2 =
2 $% ! $' " > %= R%9 % 60 H % 1 $% ! .20
R%9 % 2 % 4 1 $% ! R%9 % 5 .& - H % & ' "%=
- $ $ ! . ' .4 2 $% ! (30 /38)
43
Y 1 $% !
i) ii) ii)
.1
26 26 1 . = 52 52 4 13 13 1 . = 52 52 16
13 13 13 13 13 13 13 13 1 . + . . . = + + 52 52 52 52 52 52 52 52 4
& ' ! & . 23 = . $%! "
.2
3 J 0 %
$) * ) < # = .+ % 3
K K
1 1 1 + = 8 8 4
(1 / 2 . 1 / 2 . 1 / 2) + (1 / 2 . 1 / 2 . 1 / 2) =
44
1 4
(i
# < # = (ii 3 8
=
C2 3
8
45
=
=
3 = 8
25 = 32
5
C5 5 C4 5 C3 5 C2 5 C1 5 C0
÷ 32 = 1/32
$%! " .3
= $! " = = "
÷ 32 = 5/32 ÷ 32 = 10/32 ÷ 32 = 10/32 ÷ 32 = 5/32 ÷ 32 = 1/32
p ( S S S)
=
1 1 1 1 . . .= 2 2 2 8
.4
$ ! > " ! 3 > C : %@ 3n = $# $ % " !
.5 1/2 = $! < # = = =
< # =
"
= + " ! 3 ' > C & "
∴ n J + C ! ! ' > C & 2 n n ×1= 2 2
= + C " !
∴ = + " !
n n 5n ×3 + × 2 = 2 2 2
46
= + C $ ! ! n ÷ 3n 2 n 1 1 = × = 2 3n 6 =
:& . = $%! " .6 +
+ C
+ C
+
+
+
+ C
+ C
! ! S-% = + C -@ < # D ! !
.
9 26 3 3 1− . = 1= 5 7 35 35
+ $ %) + C < @ = + C -@ < # D ! ! + C $ %) + < @
+ C $ %) + C < @
2 3 3 4 2 4 = . + . + . 5 7 5 7 5 7 26 = 35
47
+ C " ! 3 * ) ! = 4 # .7 + 3 * ) + 3 * )
2 1 6 5 4 3 2 1 1 3 P = . . + . . . . + = 12 11 10 12 11 10 12 11 10 10 أو
=
C3 3
+
C3 12 C3 6
12
C3 i)
P
+
C3 3
$%! " .8 = 220 9
C3
( ) =
P (+ ! )
= =
iii)
P (+ -@ < # ! ) = =
48
3
12
C3
C0
84 21 = 220 55
=
ii)
1 10
=
3
C1
9
12
C3
C2
27 55
3
9
C1 C2 + C2 C1 + C3 C0 12 12 12 C3 C3 C3 3
34 55
9
3
9
= 1 -P (+ ! / ) 21 55
= 1 34 55
=
iv)
5
P ( # 0 1 % #! " ! ) =
C3
P ( # 0 1 % ! ) =
I $ - × =
=
1 100
4
3
C3 + C3 12 C3
3 44
=
v)
+
×
5
4
3
C1 . C1 . C1 12 C3
=
=
3 11 .9
50
1 2
. % = % ! 4
∴
. $- ∴ 4 # .10
P =
45 25 30 . . 100 100 100
= 0. 0337 5 49
< @ $- & ' # < # = 0 1 %
(i
1 = < @ $- 0 1 % 52
(ii
' % 5 %. @
1 /2 =
- ! = 4 #
10
.11
... 2 1
4 4 4 4 = . . + + ... 52 52 52 52 4 4 = 10 . 52 52 10 = 3 13 10 = 169 4 1 4 = 13 . = 52 52 13
8 $%! "
S =
(iii .i .12
5 $ "
5 = 4 # ∴ 8
5 0
P =
2 8
=
1 4
.ii
(1/3) = $) * ) " E ! .13
$#' 4. , E ? ( 2 4. , . 4 # P ( E ? )
E ? : 4. , E ?
1 2 3 3
11 3 4
1 3
1 6
= + + 13 36
=
1 5 × 2 13
1/2 = %=
J E .14
= 4 @ %= 4 - !
1 6 × = 4 & %) %= 4 - ! 2 17
P (4 ) =
1 5 1 6 + 2 13 2 17
= 0. 36 8 7 8
i)
P ( "%)
=
ii)
P ( % ) ! )
=
iii)
4
C1 16
4
C3
P ( ) ! @ < # "% 3 * ) ) =
1−
=
15 16
& I %) %-
.15
S = 24 = 1 6
+
16
4
C4
C44 16
4
4
4
4
C0 + C1 + C2 + C3
=
16
5 1
P (A B)
.16
= P(A) + P(B) - P(AB) = 0. 6 + 0. 3 – 0. 1 = 0. 8
.17 P(AB)
= P(A) P(B/A) = P(B) P(A/B)
∴
P(A)
=
P(AB) 1 / 5 2 = = P(B / A ) 1 / 2 5
P(B)
=
P(AB) 1 / 5 3 = = P(A / B) 1 / 3 5
! .18
M1 > %= M2 > %= M3 > %=
[ = ! = P(D)
[ = ! = P(M1) [ = ! = P(M2) [ = ! = P(M3) $ O " & $? "%
P(M1) = 0. 3
P(D/M1) = 0. 01
P(M2) = 0. 3
P(D/M2) = 0. 03
P(M3) = 0. 4
P(D/M3) = 0. 02 5 2
$ " 4% O % P(M1D) = P(M1) P(D/M1) = (0. 3) (0. 01 ) = 0. 003 P(M2D) = P(M2) P(D/M2) = (0. 3) (0. 03) = 0. 009 P(M3D) = P(M3) P(D/M3) = (0. 4) (0. 02) = 0. 008
. % # ) ! ( " D, .
$ / % E
P ( M1 / D ) =
=
P( M 2 / D) =
P(M1D) P ( M 1D ) + P ( M 2 D ) + P ( M 3 D )
0.003 3 = 0.003 + 0.009 + 0.008 20 P (N 2 D) P (M1D) + P (M 2 D) + P(M 3D)
=
0.009 9 = 0.020 20
P ( M 3 / D) =
0.008 8 = 0.020 20
5 3
:& $ " S E # !
9 /2 0
0 .0 0 9
0 .0 3
0 .3 0
M2
3 /2 0 8 /2 0
1.00
0 .0 0 3
0 .0 1
0 .0 0 8
0 .0 2
0.02 0
0 .3 0
M1
0 .4 0
M3
1.00
Q % E .19 P ( 4 2 ) =
=
4 1
c c 52 c5
4 48 4
3243 10829
& ' C #E 3 , . B - $ ' % " 0 E 5 %.
' / & < @ $ , E ? % 4 2 $- 4 4
= ! $ D, . & ' 4 4 2
4 48 47 46 45 4243 × × × × = 52 51 50 49 48 54145
. 4 # ' $#) 0 E " 3 3243 = 54145 3243 = 10829
5×
5 4
.20 P( D / M1) = 0 . 0 5 P( M) = 0 . 6
M1
P( M2) = 0 . 4
M2
D
P(M1D) = P(M1) P(D / M1)
= 0.60 × 0.05 = 0.03
N
P( D / M2) = 0 . 0 2
D
P(M 2D) = P(M 2 ) P(D / M2 )
= 0.40 × 0.02 = 0.008
N
P(M1 / D) =
P( M1D) P( M1D) + P( M 2 D) 2
0.03 0.03 + 0.008 0.03 30 = = 0.038 38 = 0.789 5 =
5 5
5 6
RANDOM
V ARI AB L E S AND P ROB AB I L I T Y DI S T RI B U T I ONS
& % . , ) 4 = 1 & ' %
H I % c ... 4 # $ - 4 D ! 4 % $ -
% $ & ' ! " % ! $ - $ I 2 4 # = ! &
" V < . 4 # $ # $ - " ! & ' ! $ % .$ I 2
(* =
DISCRETE ! ! & I 2 V
CO N TIN O U N S :
$ = @ - $ - , E ? J , & I 2 V .
... 3 2 1 )
: > V < # $ #)
D : % ! $ I = 9 $ % & ' @ '
P
c ... " " ! 2 Q M 2 & ' : "
P
Q * E & C 1 $ # . @
57
P
:
) Q E $ - J , E ? J , & I 2 V .
.$ 0
: = V < # $ # )
. @ " N Q M 4 # $ 2 "
P
.0 Q 9 & ' & I @ = 4 *
P
. $ $ #
.c ... E @ $ % * E & ! 1 % # ) D i s c r et e
P
P
P r o b a b ility D is tr ib u tio n s
! " $ % ! H I % ) % % % ! : % ; ' % D . % , 9
:& %
1
2
1 /6
3
1 /6
4
1 /6
1 /6
5
6
1 /6
1/ 6
% % ! : % ' . & B % ! % & . % , 9
:& % ! $ % ! H I % ) %
2
3
4
5
6
7
8
9
1 0
1 1
1 2
1 36
2 36
3 36
4 36
5 36
6 36
5 36
4 36
3 36
2 36
1 36
58
"
& I 2 V - C J , , .
: ; ' , ! . > & I 2 V # & > < / %
! " # # $ % & % ' ( ' ) # * + % " # . , - .
$ % , - #
Pr o b a b i l y De n s i t y Fu n c t i o n (P.D.F.)
$ 1 # E - , E ? X > V % ! , 9
xn ... x3 , x2 , x1 ,
: < , .
;'
i = 1, 2 ...n 3
P (X=xi)
. f(xi) $ ' ) ! $
x $ 0 1 - + ( 2 0 -
* - ( * x ! ( ! $
3 # ( * f(x) & , & i) f (x) ≥ 0
P(X=xi ) :
n
ii) ∑ f ( x i ) = 1 i =1
:& ! 2 X > V # $ ' ) ! : , E ? X = x
x1
x2 … … … … ..xn
f (x) = P (X= x)
f(x1)
f(x2) … … … ..f(xn)
59
: 1
= " . X > & I 2 V % ! , 9 5 5 K =
5 5 K =
K =
K =
}=
& % L 1 $ % ! "
2 1 0 & . X $ %! ,
f ( x ) = P (X = x)
f (0) = P (X = 0) = 1/4 f (1) = P (X = 1) = 1/2 f (2) = P(X = 2) = 1/4
:& # ! . $ % ! X & > ! X =
0
x
f (x) = P (X= x)
1
1 /4
2
1 /2
1 /4
:& # ! % , . !
1
{
: ; ' $ % $ - & % $ ! : < #
3/ 4 1/ 2 1/ 4 0 1
2
6 0
X
P r o b a b ility D is tr ib u tio n
$ & I 2 V , E ' % 4 V & ' % % !
- $ X V , E (C % % $ % ! $ -
& > : . % $ D , . $ % ! $ - $ - Q
. X V #
X ( ! $ X $ % & + ( - F(x) & , & F (x) = P(X ≤ x) =
P(X ≤ x)
∑
xi ≤ x
( *
* x 5 *
f ( xi)
C o n t i n u o u s P r o b a b i l i t y D is tr ib u tio n s
: % ? = & I 2 V & > % (C % % !
= V , $ 1 # E % J , $ I . D , . $ - ! S
% 0 ! ! : % ! d $ ! % ! $ W % F
E
: 2
1%
# E $ ! % ! & ' # E !
& = & I 2 V % . 0 ! ' E W < 9 0 !
: $ D , D + # 0 ! 100 &
6 1
! " # $ % & X 0 .9 0 .9 0 .9 1.0 1.0 1.0
( )
0
1 7 25 32 30 5 1 0 0
5 9
f (x)
0
1 5
0 .0 1 0 .0 7 0 .25 0 .32 0 .30 0 .0 5 1 .0 0
[ I 2 0 ! E $ - % % # '
& I 2 V , $ % ! ' 1. 05 0. 9 5 % - J
= V $ & ' $ # ! 2 H % % % ! 5 , . !
$ 1 '
! < #
.> V :
& ' . !
.$ # = $ I 2 " V : & ' " % % ! = E
! 2 & ' ! a ≤ x ≤ b J (a.b) Q & ' > (* = V X !
&
f (x ) p (c ≤ x ≤ d )
a
c
d
6 2
b
x
(a, d ) Q & ' x = & I 2 V > ' & ' @
:& # ! & P(c ≤ X ≤ d) :
f(x) $ = $ =
. ! 2 & ' $ # # / $ J x = d
P(c ≤ x ≤ d)
, x =c K
: ! , 9 X = V # $ ' ) ! $ f(x) !
i)
: f(x)
f(x) ≥ 0
∫ f (x) d x = 1
ii)
1 =
R
∫
b a
∫
∞
< % % " $ !
f (x) d x = 1 (a ≤ × ≤ b ) f (x) d x = 1 ( − ∞ ≤ × ≤ ∞ )
−∞
P( x ≤ a)
J a J - X ? & .
, ! . F (x)
F( x ) = P ( x ≤ a) = ∫ f (x) dx a
-∞
6 3
:& ! 2 & ' ! & .
f(x) f(x) F(a)
−∞
f(x)
X
x= a
:3
J ( a, b) Q & ' X >
b
P(a ≤ x ≤ b) = ∫ f ( x )dx a
=
b
∫
f ( x )dx −
−∞
a
∫
-∞
f ( x ) dx
= f (b) - f (a) : K
f(x)
F(b )-F(a)
−∞
f(x)
∞
a
b
6 4
x
& . : : & I 2 V f (x) = c
(5x )
x " ) c ! , 9 P1
x=0 , 1, … , 5
c " ) $ - K K
?'
" % ! $ % & ' 3 ) & I 2 V X ! , 9 P2 : $ = < # & . X : " ) C
X=x
0
f ( x) = P ( X=x)
C
1
2
3
4
2C 3C 4C 1.5C
5 0 .5C
C :
i) ii) iii)
P (x< 3)
: :
P (0 < x ≤ 4 )
P (0 < x < 2 )
X :
6 5
" % ! $ ! : < # = < $ % $ - " , 9 P3
< % $ - " & " ) & I 2 V X . X V : .$ ! : < # =
10 $ ( " ! ) : 2 E P4
? ' E 2 - ) X & I 2 V ! , ; ' : 2 . X K :
: $ ' ) ! & I 2 V X ! , 9 P5 f(x) =
i) ii) iii) iv )
1 2
0 ≤ x ≤ 2
P ( 0 .5 < x < 1.5 )
: K K
P ( x > 0 .25 )
P ( x < 0 .75)
P (x>3)
f ( x) =
2−x 2
0 < x < 2
6 6
" % ! , 9 P6
: K K
P ( 0 .5 < x < 1 )
i)
P ( x > 1.5 )
ii)
P ( x < 0 .3)
iii)
P ( 0 < x < 2)
iv )
! , 9 $ # I $ E ! $ 1 2 " % & ' P7 . 0. 6 . $ =
: 2 ' - $ =
: N
J < #
% 4
" 9 # 4 & $ # I @ %
. x $ M & '
. & x > 4 % , 9
K K
4 3 2 & . - $ ) * ) : # 4 P8
% C ' , 9 $ ) * ) - @ < # ! & ' 2 N 2 N 3
H % $ % ! H I % = . % - B ) x .& x >
1 . + = M & ' 4 U 4
= .
% ! , 9 P9
% ) x ? % C ' , 9 .4 * $ ) * ) *
. x K & > H % $ % ! H I %
6 7
P1
∑ f (x) = 1
a11x
∴
∑ C ( x5 ) = 1 C ∑ ( x5 ) = 1
C ( 1 + 5 + 10 + 10 + 5 + 1 ) = 1 C = 1/32
∑ f (x) = 1
: P2
a11x
i)
∴
C + 2C + 3C + 4C + 1.5C + 0 .5C = 1
∴
C = 1/ 12
P (x< 3) = P (x=0 ) + P (x =1) + P (x = 2) = C + 2C + 3C =6 C = 6 (1/ 12) = 1/ 2
ii)
P (0 < X ≤ 4) = P (X=1) + P (X=2) + P (X=3) +
P (X = 4) = 2C + 3C + 4C + 1.5C 6 8
:
= 10 .5. C = 10 .5 (1/ 12) = 10 .5/ 12
= 1 – [ (P (X=0 ) + P (X=5)] = 1 – (C + 0 .5 C) = 10 .5 C = 10 .5 12
iii)
P (0 < X <2) = P (X =1) =2C = 2/ 12 = 1/ 6 x : F(x) = P (X ≤ x) F(0 ) = P (X ≤ 0 )
= 1/ 12
F(1) = P (X ≤ 1)
= 3/ 12
F(2) = P (X ≤ 2)
= 6 / 12
F(3) = P (X ≤ 3)
= 10 / 12
F(4) = P (X ≤ 4)
= 23/ 24
F(5) = P (X ≤ 5)
=
6 9
1
P3 f(x) = (1/ 2)x x= 1, 2, …
X=x f ( x) = P ( X=x)
1
2
3… … …
… .10
1/ 10
1/ 10
1/ 10 … …
… .. 1/ 10
f(x) = 1/ 10
i)
P4
x = 1, 2, …
P (0 .5 < x < 1.5) =
∫
1.5.
1 / 2 dx
0.5
=
x 2
1.5 0.5
=½ ii)
P (x > 0 .25) =
∫
2
1 / 2 dx
0.25
= 7/ 8 iii)
P ( x < 0 .75) =
∫
0.75
1 / 2 dx
0.5
= 3/ 8
70
, 10
P5
iv )
∫
P (x > 3) =
∞ 3
=0
0 ≤x ≤ 2 1
i) P (0 5 < x < 1)
=
=
=
1 / 2 dx
∫
2−x dx 2
∫
1-
∫
x 0 dx -
0.5 1 0.5 1
x dx 2
0.5
= x
1 0.5
= 1/ 2 -
= =
iii) P (x < 0 .3) =
∫
0 .3 0
∫
0.5.
-
x2 1 4 0.5
3 5 = 16 16
∫
2
ii) P (x > 1.5)
1
2−x dx 2 1.5
1 16
2−x dx 2
= 0 .2775
71
x dx 2
P6
∫
2
iv ) P (0 < x <2) =
0
2−x dx 2
= 1
x
0
p(x)
0.4
P7
1
2
( 0.6 )
( 0.6 )
1
( 0.4)
3 ( 0.6 )
2
( 0.4)
( 0.4)
4 3
( 0.6 )
5 ( 0.6 )
4
5
( 0.4)
P8 x p(x)
4 1 9
5 2 9
6 3 9
7 2 9
8 1 9 P9
P (x) = Cnx p P (0 ) =
27 64
P (3) =
1 64
x
q
n-x
, P (1) =
72
27 64
, P (2) =
9 64
73
"1 =
% ( " ) "# F !
. F C : I = " % &I = M > 0 . > # > C ! . : "# D , .
. V * 0 . 2 % : "# D , . $ % % &# '
Expectation
' f(x) : $ ' ) ! &I 2 V x ! , 9
∫ x f(x) dx
∞
-∞
< # : % ! ( ! : % ) ( ) ! , . . > -
f(x) $ ' ) ! $ % C & " )
J $ D , . &' > - U = x V ! , 9
∑
all x
x P(x)
x V # $ $ P(x) 3
$ C $ # < 9 2 E 3
E(X) x > - %
: % C x V < # Q $ %
74
E (x) =
∫xf
∞
(x) dx = µ
-∞
=
∑ xP(x)
=µ
all x
.( * = x V ! 4
E & > # & E(x) > - < ! . x V # $ - $ # 4 V &' ( ) µ & M
: 2 % ( (* ) > V $ &' ) > - $ V =
/ *
4 > - ( i = 0 1 , 2) xi 1 % % $ # P(x) (%
. : % &. @ D , . ) !
&. ( ϕ (x) (* ) % ) x &' $ J ' ( I 2 ( V x ! , 9
:J : - &I 2 V Q E @
∑ ϕ (x) P(x)
E [ ϕ (x)] =
=
all x
∫ϕ
Range x
(x) f(x) dx
. (* = ( x ! 4 , K K E ?%
ϕ (x) = x r
E (xr) =
75
∑ xr
all x
P (x)
K K !
=
∫x
r
Rx
f (x) dx
%
r = 1 E (X) = =
∑x
all x
K K !
P (x) = µ
∫ x f (x) dx
Rx
=µ
: 1
&C > - . ' $ " 3 * ) $ % % $ - "
Y" = " / "
:& &' / % " &% L 1 ) % % % !
1 /8
S
S
S
5 S
3
1 /8
5 S
2
1 /8
2
1 /8
5 5 5
0
1 /8
S
5 5
S
S
1
1 /8
1
1 /8
2
76
S
5 5 S
1
1 /8
S
5 S
5
5
B X " % D =
: < # = " )
V # : " 3 * ) % : - X :& &' D X=x
0 1 8
f ( x) = P ( X = x) 3
∑
E(X) =
i=0
2
3 8
3 8
3 1 8
Xi P (xi)
1 8
=
1
3 8
3 8
1 8
0 ( ) + 1 ( ) + 2( ) + 3( ) 3 2
=
&% &' $ % $ $ ) % % % !
:&
f ( x ) = P ( X= x ) 3/8 2/8
• • • 0 1
2
77
3
X
S
P r o p e r t i e s o f E xp e c t e d V al u e
> - % E : % $ C $ # < 9 2
E , . , & &I = M V < # J ! E (ax) =
aE (X)
∑ ax P(x)
E (ax) =
:$
x a
:K K K .
all x
a ∑ X P(x)
=
all x
=
a E (X)
∫ a xf (x) dx
∞
E (ax) =
-∞
=
a
∫ f (x) dx
∞
-∞
=
a E (x)
! , 9 a ") X ; ' &I 2 V E (a) = a
78
E (a) =
∑ aP(x)
all x
= a =
=
: K K K .
∑ P (x)
a (1)
a
E (ax+ b ) = a E (x) + b
:$K K K
%
: K K K .
E(ax+ b ) = E (ax) + E(b ) = a E (x) + b : 2
E(2x + 3) = 2 E(x) + 3 # " $ % E [ E(x)] = E (x)
# " # " :K K K .
") J ") > - % !, ! $ ) $ - > - .> - J > - > - ,
' E(z) = z
> - < 9 % #' E(x) $ ) $ z
% E[ E(x)] = E (x) 79
&4E [ ( X - E ( x) ] = 0 E [ (x – E (x) =
: K K K .
E (x) - E(E (x)]
= E (x) - E (x) = 0 :
Y X
E ( X ± Y) = E ( x ) ± E ( Y)
: 1
E (X ± Y)
∫∫
∞
=
:K K .
(x ± y) f(x.y) d xdy
−∞
= = = =
∫ ∫ x f (x, y) dxdy ± ∫ ∫ y f (x, y) dxdy ∫ x [ ∫ f(x, y)dy]dx ± ∫ y [∫ f (x, y) dx]dy ∫ x f(x) dx ± ∫ y f(y) dy E(X) ± E(y )
: ! "
Y, X
E ( XY) = E ( X) E ( Y) E(X Y)
=
∫ ∫ X Y f (x, y) d x d y 8 0
: 2 : K K K .
∫ ∫ xy f (x) f(y) d x d y … … ∫ x f (x) d x ∫ y f (y) d y
= =
=
. *
E (X) E (Y) : 3
:&. : : I 2 V X ! , 9 f(x) = 2x
0≤X≤1
E(x+ 1)2 E(x)
E(x2)
∫x
f (x) dx
∫x
(2x) dx
1
=
0
1
=
0
∫ 2x 1
=
2
dx
0
= = =
2
x3 3
2
1 -0 3
1 0
2 3
8 1
K K ! K K E (x)
?' :
∫x 1
E(x ) = 2
2
0
∫
1
=
0
f (x) dx
x 3 (2x) dx
∫ 2x 1
=
3
dx
0
=
2
x4 4
1 0
1 2
=
E [ x2 + 2 × + 1]
E (x + 1)2 =
E(x2) + 2 E (x) + 1 = =
½ + 2 (2/ 3) + 1 = =
½ + 4 /3+ 1 17 6
Variance & Standard Deviation " K 2 # 0
K &K . J K K % K %
" K 2 # 0
! O 2 - % % % % ! J ! > (D is p e r s io n ) . &I 2 V & > #
K V &K > K ( &K ) : - $
K $ " # % & > ! < # % &I 2 8 2
% ! ) ! $ + 9 &' , 9 Q &'
: K &I 2 V - " 2 2 % Q " # J %
0
K K K ( 2 0
) ! < 9 % . % .Q E < 9 .J % &. &I 2 V # & > " 2
$ K 0 1 % K 0 &I 2 V & >
> % O % % . 1 J ! > $ &' : -
K "' % > 9 $ - $ "' % .J ! > $ % # &
-
X
X ! ! " " # $ % & µ E ( x ) =
' % ( X 2
σx
2
σx
= V a r (x ) =
) V a r ( x )
∑ (x − µ) 2 p (x)
all x
∫ (x − µ)
∞
=
-∞
2
f (x) dx
+, - ) # $ * = E [ ( x -u ) 2]
= E [ x - E ( x ) ]2
8 3
X
. - "' % > > - . J
- / * % X K $ &. 4 &' $ #E X , !.
- 0 1 % " 2 0 ' X < # = % &! $ % 9 - 0 1 % " 2 # 0 X . # 4
& , % % % ? '
.J % <
-
X % . * . X / σ x : σx
=
σ 2x =
E (x - µ ) 2
J K % 4 < 9 4 * U = % % .+
, . &' - &C - J ! > # : 4
B " K 3 * ) % : - + 9 . (1) ) <9 x & K .
$ K & K & . =
: / " & I 2 V
V # x ! 4 $ 1 ! σ 2x J K %
σx .
8 4
X = x
f ( x) = P ( X=x)
Xf ( x)
1
3/ 8
3/ 8
( 1-3/ 2) 2 = 1/ 4
1/ 8
3/ 8
( 3-3/ 2) = 9 / 4
0 2
1/ 8
0
3
3/ 8
6 /8 E (x ) 12 3 = 8 2
( x-E ( x) )
[ x-E ( x) ] 2 f ( x)
2
( 0-3/ 2) 2 = 9 / 4
9 / 32 3/ 32
( 2-3/ 2) = 1/ 4
3/ 32
2
9 / 32
2
σ 2x =
σ 2x =
J % σ x = = E
(x) =
µ
3 4
24 3 = 32 4
?' , ! .
3 4 3 2
X :
V a r (x) = σ 2X
σ2
=
E (x2) - [ E (x)] =
σ2
2
E (x2) - µ 2
: K K K .
= E [(x - E (x)]2
= E [(x2 – 2xE(x) + {E( x )}2 ] = E (x2) – 2 {E( x)}2 + {E( x )}2
= E (x2) - [E ( x )]2
85
: 5
K
K ' " 3 % : - & . ) <9
$ / % E x K J % 4 : & &
$
[ σ 2 = E (x2) - µ 2
X = x
f(x)
xf(x)
x2f (x)
0
1/ 8
0
(0)2 = (1/ 8 ) = 0
1
3/ 8
3/ 8
(1)2 (3/ 8 ) = 3/ 8
2
3/ 8
6 /8
(2)2 (3/ 8 ) = 12/ 8
3
1/ 8
1/ 8
(3)2 (1/ 8 )= 9 / 8 24 E(x2) = =3 8
E(x) =
σ2
3 2
J ' , ! .
= E(x2) - [E (x)]2 3 = 3 - ( )2 2 3 = 2
= # 4
& , J J % σ=
86
3 3 = 4 2
Properties of Variance
J @ 4 % # S
E D , . # S
:& . S
E D 5 % .
E . X & I 2 V & ' :
X a V ar (ax) = a2 V ar (x)
(1
: K K K .
V ar (ax) = E [ax – E (ax)]2 = E [ax – a E (x)]2 = a2E [x –E (x)]2 = a2v ar (x) : 6
:$ " V . ' 0 . 5 J x ! , 9 2X X 2 i)
V ar (2x) = 4 V ar (x) = 4 (0. 5 ) = 2
87
(i
(ii
:
ii)
X 1 V ar (x) V ar ( ) = 2 4 1 (0. 5 ) = 4 = 0. 125
' " ) a ! , 9
V ar (a) = 0
(2
: K K K .
V ar (a) = E [a –E (a)]2 = E (a – a)2 = E (02) = 0 a) = V ar (x) ± V ar (x
:$K K K
%
: K K K .
V ar (a) ± a) = V ar (x) ± V ar (x 0 ± = V ar (x) = V ar (x)
K V <9 : ) $ - J [ $ - J $ ' C 9 ' , ! .
. V , <# )N & I 2 88
: ! . ' 5 J x V ! , 9 : 7 i)
x+3
ii)
x–6
i)
V ar (x + 3) = V ar (x) = 5
ii)
V ar (x – 6 ) = V ar (x) = 5
:
: " # Y ! X (3 V ar (x+Y) = V ar (X) + V ar (Y) V ar (x - y) = V ar (x) + V ar (y)
B J % 1 # I 2 V B J . %
: K K K . V ar (X ± Y) = E [(X ± Y) – E (X ± Y)]2 = E [ {X − E ( x )} ± {Y - E(y)}]2 = E [ {X − E ( x )}2 + {Y - E(y)}2 ± 2{X − E ( x )}{Y − E( y )}] = E {X − E ( x )}2 + E{Y - E(y)}2 ± 2E [{X − E ( x )}{Y − E ( y )}] = V ar ( x ) + Var (Y) ± 2E [{X − E( x )}{Y − E( Y)}]
89
# ) 1 =
J E @ ? " )% % # ' , ! . ( E !
= 2 [ E {x − E( x )} E{Y - E(y)}] = 2 [(0) (0)] = 0
∴ V ar (X ± Y) = V ar (x) + V ar (Y) J : # " V
J <# D D , . ! :
V ar (x1 ± x2
± x3 ± …
± xn)=V ar (x1)+V ar (x2)+… + V ar (xn) C o variance
$ ' )! : f(xy) " % ! , ' Y, X V * 0 . : & # ! ( C
Co v (X, Y)
=
Y, X V V ' V , .
∫∫
∞
[X –E (x)] [Y – E(y)] f (xy) d xd y
−∞
= E [ {X − E( x )}{Y - E(y)} ]
9 0
P
C ) Y, X
! " # $ % $ ( Co v : & ' $
Co v ( x , Y) = E [ ( x - µ x) ( Y- µ y) ]
X
X, X 1
Co v (x, x) = V ar (x)
: K K K .
4 # <9 = % D * V : & ' Y X > C :
b ,a
Co v (ax, b y) = ab Co v (x, y) Co v (ax, b y) = E [{ax − E(ax)}{by − E(by )}]
2 : K K K .
= ab E {x − E( x )}{Y − E( y )} = ab c o v (x, y) Co v (x, a) = 0
9 1
a 3
Co v (x, a)
= E [{x − E( x )}{a − E(a )}]
: K K K .
= E [{x − E( x )}{0}] =0
P4
Co v (x1 + x2 , y) = Co v (x1, y) + Co v (x2, y)
: K K K .
Co v (x1 + x2 , y) = E [{( x1 + x 2 ) − E ( x1 + x 2 )}{Y - E (y)}] =E [{x1 − E( x1 )}{y − E( y )}] + E [{x 2 − E( x 2 )}{y − E ( y )}]
= Co v (x1 , y) + Co v (x2 , y)
x2 , x1 5
V ar (x1 + x2) = V ar (x1) + V ar (x2) + 2Co v (x1 , x2) V ar (x1 - x2) = V ar (x1) + V ar (x2) - 2Co v (x1 , x2)
: K K K .
V ar (x1 + x2) = E [ (x1 +x2)
- E (x1 +x2)]2
= E [{x1 − E( x1 )} + {x 2 − E( x 2 )}]2
= E [x1 − E( x1 )]2 + E[x 2 − E ( x 2 )]2 + 2 E [{x1 − E( x1 )} {x 2 − E ( x 2 )}]
= V ar (x1) + V ar (x2) + 2 Co v (x1 , x2)
: & ' J V 1 : & ' ) 9 2
V ar (x1 – x2)
x2 , x1 6
Co v (x1 , x2) = 0 Co v (x1, x2) = E [{x1 − E( x1 )} {x 2 − E ( x 2 )}]
: K K K .
= E[x1 x2 + E (x1) E(x2) – x1 E (x2) – x2 E(x1) = E(x1) E(x2)+E(x1) E(x2)–E(x1) E(x2)-E(x1) E(x2) = 0
! # V E (x1 x2) = E (x1) E (x2)
Cov (X , Y) = ρ Var (x) Var (y)
. 1 =
Y,X
J Y , X V ! , 9 1 =
, . !
Q & ' >
1 ≥ ρ ≥ -1 ρ2 ≤ 1 93
;K ' , K ! .
: " )? ' E (x) = µ ! , 9 (1
i) E (x- µ ) = 0 ii) E (x-c )2 = E (x - µ )2 + ( µ -c )2
5 4 3 K = E 2 S
. " ) J C 3
E 2 " " % ! , 9 (2
: 4 <# & . I # < @ $ & ' S
0. 04 0. 09 0. 12 0. 27 0. 43 0. 05
Y I # < @ : & ' . > - S
E 2 6
E 2 @
. '
a " ) ! % Q V = : % & ' ! E (x-a)2 " ) (3 . E(x) J
:& . $ ' )!
Y , X # V % ! , 9 (4
1≤ X ≤ 0
f (x) = 12 x 2 (1 – x)
1≤ Y ≤ 0
f (y) = 2 Y
Y
x
2
+
94
x Y
> -
$ - 6 : % 10 & : & I 2 V X ! , 9 (5 E (x2 + 3 x)
- $ "
& > & & (6 : 2 * E " & I
! ! "
0 10 20 30 40
0. 01 0. 05 0. 39 0. 45 0. 10 1.00
# $
. 2 * E : "
. 2 * E : "
: KK
(i
(ii
" * & ' : : % # " % 1 #
& P7 : % ! / % (
% & ' ! ! "
0 4 6 8 40
0. 13 0.27 0. 39 0. 21 0. 07 1.00
# $
95
$ @ "
: KK
(i
$ @ " # & (ii & K. < K @ I U > 4 = % $ - S
E 2 J 2 (8
0. 005 K K% 3000 & K. $ K % ) I % 4000 Y $ - D , 4 % . ' 4 < # 0. 008
C B A K E K K! , K' D + 2 S
E 2 4 . ,
2000 A K K ) ! 4 < # 0. 3 0. 3 0. 4
a K# . ' %
(9
3000 C % 25 00 B % YS
E 2 , . : ' > -
: K - " I C , B, A & . $ # > 2 $ ) * ) & ' $ ! 2 (10
4 K < K#
25 . 000 5 0. 000 100. 000 & K. $ ) * ) > 2 #
: . ' 4 < # % 3.000 5 000 10. 000 & . " % . $ ) * ) > 2 $ - " I &
. &
$ 1 =
(i
(ii
& ' $ + E @
& . X " % ! , 9 (11 f (0) = 0.9
f (1) = 0.05
f (2) = 0.03
f (3) = 0.02 96
Y $ 1=
200 & ' > - + E @
. '
: : & I 2 V X " ) C ! , 9 (12 f (x) = c x x = 3, 4, 5, 6 C " ) $ -
X > -
X
: KK
(i
(ii
(iii " ) (13
Co v (x, y) = E (x y) - µ x µ y & .
y, x V # $ ! 2 $ ' ) ! : " # , 9 (14 1≤ X ≤ 0
f (x, y) = x + y
1≤ X ≤ 0 Co v (x, y) J y, x V
(x, y) J y , x
97
KK
(i
(ii
(1 i)
E (x- µ ) = E[ (x-E (x)] = E (x) – E (x) = 0
ii)
E(x-C)2 = E (x2-2c x + C2)
@ P
= E (x2) – 2CE(x) + C2 = E (x2) – 2C µ + C2
@ P
E (x- µ )2 + ( µ -c )2 = E (x2 - 2 µ x + µ 2) + µ 2- 2 µ C + C2 = E (x)2 – 2 µ 2 + µ 2 + µ 2 - 2 µ C + C2 = E (x2) – 2 µ c + C2
4 # . ' ! ∴
X = x 1 2 3 4 5 6
f(x ) 0.05 0.43 0.27 0.12 0.09 0.04 ∑ XP( x ) E(X) =
allx
98
x f(x ) 0.05 0.86 0.81 0.48 0.45 0.25
(2
= 2.89
4 # . J : < @ : 2 " % ! , 9 Q V = : % & ' ! (3 1=
E ( x − a ) 2 = E ( x 2 ) − 2 a E (x) + a 2 144424443 Z
) , . 2 % Z K $ % (a
dz = 0-2 E (x) + 2a da = 0 – 2 a + 2a =0
a = E (x) 3
Q V = : % & ' ∴
Y X Y X E 2 + = E 2 + E Y Y X X
(4
1 1 = E (Y) E 2 + E(X )E Y X
D * : & ' H % F
99
* %@
% D < # > - ! % O
E (Y) = =
∫
1
∫
1
Y f (Y) d y
0
Y (2Y) d y
0
= 2 ∫ Y2 d Y 1 0
Y3 3
= 2 =
1 E 2 X
= =
1 0
2 3
∫
1
1
{12 X2 (1 - X) } d X
1
{12 X2 - 12 X3 ) } d x
X
0
∫
1
X
0
2
2
= 12 ∫ (1-X) d X 1 0
= 12
X-
1 = 12 2 = 6
10 0
X2 2
1 0
E (X)
∫
1
=
0
{
}
X 12 X 2 (1 - X) d X
= 12 ∫ X3 – X4 d x 1 0
= 12
X 4 X5 4 5
1 0
1 12 3 = 12 1 - = = 4 5 20 5 1 E Y
= =
∫
0
1 (2Y) d y Y
∫
2d y
1
1 0
= 2
Y X 1 1 E 2 + = E (Y) E + E(X) E Y X2 Y X 2 3 = (6) + (2) 3 5 = 5.2
10 1
(5 E (X2 + 3X) = E (X2) + 3 E (X)
V ar (x) = E (X2) - {E(X )}2 6
… … …
1
… … …
2
… … …
3
= E (X2) – (10)2
E(X2) = 106
1 3
E (X2 + 3 X) = 106 + 3(10) = 106 + 30 = 136
X–x 0 10 20 30 40 E(x)
=
(6 f ( x ) = P ( X= x )
Xf ( x )
X2 f ( x )
0.01 0.05 0.039 0.45 0.10
0 0.50 7 .80 13.50 4.00
0 5 156 405 160
∑ Xp( x )
E(x2) =
allx
= 25.8
∑ x 2 P( x )
allx
= 7 26
10 2
V (x) = E(x2) - {E ( x )}2 = 7 26 – 665.64 = 60.36 (7 X
P ( X)
XP ( X)
X2 P ( x )
2
0.13
0.26
0.52
6
0.32
1.9 2
11.52
0.7 0
7 .00
4
0.27
8
0.21
10 E(x)
1.08 1.68
0.07 =
4.32
∑ XP(x )
13.44
E(x2) =
= 5.64
∑ x 2 p( x )
= 36.80
V (x) = E(x2) - {E ( x )}2 = 36.80 – 31.809 6 = 4.9 9 04
≅ 5
10 3
E(x)
=
∑
allx
X P (X)
(8
= 4000 (0.005) + 3000 (0.008) = 20 + 24 = 44
E(x)
=
! (9
∑ XP( x )
allx
= 2000 (0.4) + 2500(0.3) + 3000(0.3) = 800 + 7 50 + 9 00 = 2450 " # $ % & ' ( ! (10 i) E ( X1 + X2 + X3) = E (X1) + E(X2) + (X3) = 100000 + 50000 + 25000 = 17 5000 ii) V (X1 + X2 + X3)
= V (X1) + V (X2) + V (X3) = 10000 + 5000 + 3000 = 18000
10 4
E (x)
=
= ) * * + , - . / 0 (11
∑
allx
X P (X)
= 0 (0.9 ) + 1 (0.05)+2 (0.03) + 3 (0.02) = 0 + 0.05 + 0.06 + 0.06 = 0.17
* +,
2 0 0 " 1 $ * +,
2 0 0 - . / 0 ' 2 * 3
= 200 (0.17 ) = 34
i)
∑
allx
(12
f (x) = 1
∴ ∑C X =1 C
∑X
=1
C (3 + 4 + 5 + 6) = 1 1 ∴ C= 18 ii)
E (x) =
∑ X P(X)
allx
=
∑X
=
1 ( 9 + 16 + 25 + 36) 18
CX
= C∑ x 2
10 5
=
43 9
iii) V (X) = E (X2) - {E ( x )}2 = E(X2) =
= C∑X
3
∑ X 2 P (X)
allx
1 (27 + 64 + 125 + 216) 18 216 = 9 =
216 43 V ar (x) = - 9 9 1944 − 1849 = 81 95 = 81
2
Co v (x, Y) = E (XY) - µ x µ y Co v (x, y)
= E [ (x - µ x ) (y - µ y )]
(13 :
= E (xy - x µ y - µ x y + µ x µ y )
= E (xy) - µ x µ y - µ x µ y + µ x µ y = E (XY) - µ x µ y
10 6
(14 Co v (x, y) = E(xy) – E(x) E(y)
. XY Y
X
∴
E(x+y) ! $ E (y) , E(x) ! " #
E (x + y)
= = =
= =
∫∫
1 0
∫∫
1 0
1
(x+y) f (x, y) d x d y x f (x, y) d x d y + 1
∫ [x ∫ f ( xy) dy] dx
014 04 42444 3 E( x )
∫ [x ∫ (x + y ) dy]
1
1
0
0
∫ x xy +
1 0
y2 2
1
1 = ∫ x x + dx 2 424 3 0 1 1
f (x)
1
0140442444 3 E ( y)
d y +
+ 1
=
7 7 + 12 12
1 0
1
0
0
∫y
1 0
x2 + yx 2
1
424 3 0 1 f (x )
y3 y 2 + + 3 4
10 7
∫ y [ ∫ ( x + y ) dx]
1
∫ y y + 2 dy
x = ∫ x 2 + dx + ∫ y 2 + 2 0 0 x3 x 2 = + 3 4
1
∫ y [ ∫ f ( xy) dx] dy
+
1
x f (xy) d x d y
0
1
d y +
0
∫∫
1
y dy 2
1 0
1 0
d y
d y
7 12 7 E (Y) = 12
E (x) =
1 2 1 = y+ 2
f (x)
= x+
f (y)
E (xy) = = = =
xy f (xy) d xd y
0
xy (x+y) d xd y
∫∫
x2y + xy2 d xd y
0 1 0
2 ∫ [∫ x y dx] dy +
∫ [∫
1 1
1 1
0 0
0 0
∫
1
x 3y 3
0
1 0
dy +
∫
1
0
∫
1 1 ydy + ∫ xdx 3 3 0
1 3
1
0
=
∫∫
∫∫
1
1
=
=
=
1
E(xy)
xy 2 dx] dy
xy 3 3
1 0
dy
1
∫
0
1 1 Y2 ydy + ∫ xdx = 30 3 2 1
1 1 1 + = 6 6 3 108
1 0
+
1 X2 3 2
1 0
C o v (x, y)
=
E (xy) – E (x) E (y) 1 7 7 + . 3 12 12 1 49 − 3 144 1 144
= = =
V (y) , v (x) ( ii V (x) =
E (x2) - {E ( x )}2
E (x2) =
∫
x 2f ( x ) dx
∫
1 x 2 ( x + ) dx 2
1
0
=
0
X 4 X3 + 4 6 = =
V a r (x) =
1
1 0
5 12
5 7 11 - = 12 12 144 2
109
V (y) = ρ( x , y ) =
11 144
Cov (x, y) − 1 / 144 − 1 / 144 1 == = 11 / 144 11 V ( x ) V(y) 11 11 . 144 144
110
111
THE DISCRETE PROBABILITY DISTRIBUTIONS
! " # $ ! " % & " ' ( ) *
! " # , ! " # $ + , ! " # $ + ,
! " -
. * $ # ! " ) %
$ & " . % / ! " $ + ,
. - 0 '
@ @The Binomial Distribution@
, - 0 & , $
2 ' ( ( * % $ & 3
3 , # + , & 3 4 & 3 5 . * , # 6
. 7 ! - % $ + % 8 - . 9
, + , + , & 3 ' ( $ 7 $ < . ; 3 . 9 100 :
: = 6 + ,
. .1
. .2
112
! " .3 .$
. #
% & ' % ( ) .4
! * + , - ' ) ! " .
/ ) % ( 0 " ( ! ! 0 " ! 0 1
. ! 0 " * +
:1
! 2 ( 0 " ( ! ! n 7 8
3 9 8 ! P ( % 3 ( % 4 5 6
3 & = / ! q + & + (1-P) ( ; ) : ! 3 3 & = ! ; ( ; ) 7 8
n = 1 :
( ( ?
S = (@ A 7 8
7 8
)
3 9 6 + A
3 3 9 6 + @
C
3 & = ! 0 % 1 " X D
1 E 0 ! X 2 ( /
7 8 & = ! P
( & = ! q= (I-P)
:! 9 x 1 ! * + 113
1
n=
1
X = x
f (x ) = P (X = x ) f (0) =
0
P ( ) = q
1
f (1) = P () = P
q+ P = 1
n = 2 :
( D
S = (@ @ E A F 8
@ E @ A
( " F 8
E A
9 G ! % A
A
)
A
( ( " ( 9 G ! % @ @
C
( ( " F 8
F 8
9 G ! % @ A
( " ( 9 G ! % A
@
4 ( ;
! 7 H
qq , qp, pq, pp
( ) C
2 E 1 E 0 ! ( x / 4 114
:! ! x 2 ! * + (2)
n= 2
X = x 0 1
f(x ) = P (X = x ) f ( 0) =
P ( ) = P () P () = q q = q
f ( 1) = P ( ) + P ( ) = P () P () + P () P () = P q + q P
2
2
= 2q P
f ( 2) = P ( ) + P () P () = P P = P
q2 +
2 qP + P
2
=
( q+ P )
2
2
=
1
3 n :
( ( ?
S=
{
}
8 = ( D ;
! 7 H & F = qqq, qqP, qPq, qPP, Pqq, PqP, PPq, PPP
3 E 2 E 1 E 0 ! ( x / 4
:! ! x 2 ! * + 115
3
n= 3
X = x
f(x ) = P (X = x )
0
P ( ) = P () P () P () =
f ( 0) =
q
3
f ( 1) = P ( ) + P ( ) + P ( )
1
= q
2
p + q
2
+ q p
2
p + q
p = 3 q
2
+ q p 2= 3 q p
2
2
p
f ( 2) = P ( ) + P ( ) + P ( )
2
= q p
2
3
f ( 3 ) = P ( ) =
q 3 + 3 q 2p +
P () P () P () = p
3 qp
2
+
P
3
=
3
( q+ p ) 3 = 1
: ( ( ) ( " . " = .
2= 4= 8=
21 =
22 =
23 =
( 1 ; ) n H 6
( 2 ; ) n H 6
( 3 ; ) n H 6
2 ( 2 0 " ( ( ( 7 H / %
! 2 2 ) H 6 . n 3 ( H 6 2n ; ) I 2 , 2 * 2 ) ( F 8 n F +
2 ) ( ) % ( ) ( " . " ! ( .@ H & ) ( ( 3 ( 116
F x F 8
3 9 8 J * ) #
(3 2 ( % 4 5 ) n ( 0 " ( ! ' D 2 4 2 ( ( % ( F 8 K
! @ H F (n-x) ( 4 / F x (F 8
3 & = ) C .(
Px F x (F 8 ) C D 4 qn-x F
n-x C D 4 /
px F x C D 4 K '
F 4 qn-x
; ) , 3 ) C D 4 (
C F x 7 8
E 0 7 8
n x
n n! = = x x! (n - x) !
; n x ,
3 9 8 K H
(nx )p
x
: 9 6 L ) n '
qn-x
3 & = 3 ; ) n E L. 2 2 "
: ! 9 ! 3 E 2 E 1
(33) p3 , (23) p2 q , (13) pq2 , (03) q3 p3 , 3p2q ,
3pq2
, q3
(3) ! & 8 ! I ! 117
(nx )p
( / = .
qn-x
x
x = 0, 1, …
.., n
x
(q+ p)n (q+ p)n = qn +
(1n )pq
(n2 )p q (nn−1 )p q +
n-1
+
2 n-2
n-1
+ … +
pn
(nx )p q
x n-x
+ …
+
! 0 " * + ! * + H ( ) 5 : ! 0 " * + 8 :
% x ! " # $
* " + , n & ' ( " ) # - + % $
(nx ) p
f ( x) =
x
qn-x
x= 0, 1, …, n
" + , 0 & ' $ p
" + , 0 & ! 1 $ q p +
118
q= 1
(i
(ii
(iii (iv
:.
: 2
! * ' ( ) ( ) ! ) ! K / % H 6 .( ) ) ) G ( 8 $
% ( ) ( M
: % ( M ) ) % 10 ) ; 9 ;
) / & "
) / & " "
(i
(ii
(iii
( 1 7 H ! ) " x D : 20 E ... E 2 E 1 E 0 ! x 2 ( / ) ; ) p D
) ; ) q ∴
i)
p (
p=
0. 10
q=
1-P
n=
20
! " # # $ ) = f (0)
f (x) =
f (0) = =
(nx )p q (200 ) p q x
= 0. 9 0
n-x
0
20
(q)20 = (0. 9 0)20
= 0. 122 119
ii)
P ( ) = f (2) f (2)
(202 ) p
=
=
2
q18
19 0 (0. 01)2 (0. 9 0)18
= 19 0 (0. 01) (0. 9 )18 = 0. 28 5 iii)
P ( ) =
≤ 2)
= = = =
p (x > 2) = 1- p (x 1 - {p (x = 0) + p (x = 1) + p (x = 2 )}
{
}
1 - (0.9) 20 + 20 (0.1) (0.9)19 + 0.285
1 - { 0.122 + 0.270 + 0.285} 0. 323
! 0 " * + * 4
F 2 n ( 2 ( % 4 5 6 ( , ) " 9 6 F %
x ! 0 % 1 ! * + 1 ; ) n % * 2 + H & * 4 (1) ! 9 % F 8
3 & =
2 2 2 2 2 2 2 2 2 ! 2 2 2 2 2 2 2 2 2 K 2 2 2 2 2 2 2 2 2 2 )
:! (4 )
12 0
(4)
n= 1
X 0
f (x ) q
x f (x ) 0
1
p
p
1
µ = ∑ xf ( x ) = p
0
x ! 0 % 1 ! * + ? 2 ; ) n %
2 ) ) * 2 4 (2) ! 9 % 7 8
3 & =
:! (5) ! K ) * + H & (! ) (5)
n= 2
X 0 1 2
f(x )
q 2pq p2 2
x f (x ) 0 2qp 2p2
E ( x )= 2 p ( q + p ) = 2 p
x ! 0 % 1 ! * + ? 3 ; ) n %
) * 2 4 (3) 2 ! 2 9 % F 8
3 & =
:! (6) ! 3 ) * + H & (! ) ) 12 1
(6)
n= 3
X
f(x )
Xf ( x )
1
3p2q
3q2 p
3
P3
0
q3
2
0
3qp2
6 q p2 3p3
µ = E ( x )= 3 p ( q
2
+ 2 q p + p 2)
= 3 p ( q + p )2 = 3 p
! 0 " * + (* 4 ) ! ) ) J = . H 1 ; ) n H 6 P
:
2 ; ) n H 6 2p 3 ; ) n H 6 3p
! 0 " * + * 4 J ( 7 H / % #
. n 2 n p ; )
q p #$ n % ! "
#$
µ = np
12 2
! * 4 % / ) L ' @ H µ = E (x)
n
∑ x f (x)
=
x =0 n
∑ x (nx ) p x q n - x
=
x =0 n
x n! px qn-x x ! (n x) ! x =0
∑
=
n
n! p x q (n -1) - (x -1) (x - 1) ! [(n - 1) - (x - 1)] ! x =1
∑
=
(n - 1) ! p x -1 q (n -1) - (x -1) (x - 1) ! [(n - 1) - (x - 1)] ! x =1 n
np ∑
=
n
np ∑ ( nx -−11 ) P x -1 q (n -1)- (x -1)
= =
x =1
=
np (q+ p)n-1 np
( 2 % 4 F 100 ( ( % 4 H 6 : 3 & O F 8
3 & =
3 2 2 & = ! 0 % 1 " x D : & & 2 ! 0 " * + & x ? H F 8 P=
1 2
n = 100 12 3
np = ! 0 " * + * 4
1 µ = E (x) = np = 100 = 5 0 2
K 2 2 2
# o
/ ) F n ( % 4 5 6 ( (1) " 9 6 F %
3 E 2 E 1 / 2 H 2 J n 2 % σ2x = ∑ (x - µ) 2 f(x) 4 . ! 0 " * + ( ( 1 8
9 6 8 * )
x ! 0 2 % 1 ! * + 1 ; ) n %
H 2 * 4 (1) ! 9 % F 8
3 & =
2 * + H p (4) ! K ) * + :! (7) ! 3 )
(7)
n = 1
X 0 1
f(x ) q p
x -µ 0-p 1-p
124
σ 2x =
( x - µ )2 f ( x ) q2 q (1-p)2 p p 2q
+
q 2p
= pq ( p+ q ) = pq
x ! 0 2 % 1 ! * + ? 2 ; ) n %
H 2 * 4
(2 ) ! 9 % F 8
3 & =
2 * + H (5) ! K )
2p * +
:! (8 ) ! 3 6 8
(8)
n=2
f(x )
x –
0
q2
-2p
4 p2 q2
1
2pq
1-2p
(1-2p)2 2pq
2
p2
2-2p
4 (1-p)2 p2
(&
( x -u)
2
X
u
f(x )
σ 2x = 4
2 2 p q+
( 1 -2 p) 2 2 pq + 4 ( 1 -p) 2 p2
= 4
2 2 p q+
( 1 -4 p+ 4 p2) 2 pq + 4 q
= 8
2 2 p q+
2 pq - 8 p2q + 8 p2 q
= 8 p2 q ( q -1 + p) + 2 p q = 8 p2 q ( 0 ) + 2 p q = 2 pq
125
2
p2
x ! 0 2 % 1 ! * + 3 ; ) n %
H 2 * 2 4 (3) ! 9 % F 8
3 & =
( ) ( I 3p (6) ! K ) * + 3 ; ) n 3pq ; ) * + H 9 6 8 :; ) ! 0 " * + H 1 ; ) n
pq
2 ; ) n 2pq
3 ; ) n 3P q
; 2 ) ! 0 " * + ( 7 H / %
n 2 npq
#
q p
# $ n & ) # "
#$
! * 4 4 / ) ' @ H 3
σ 2x = E (x2) – [ E (x)] E (x2) = =
2
n
∑ x 2 f (x)
x =0 n
∑ x 2 ( nx ) p x q n - x
x =0
126
x2 n ! px qn-x = ∑ x ! (n - x) ! x =0 n
= = =
n
∑
x =0
x(x-1)+ x 2 x2 $
[x(x - 1) + x] n ! x n − x p q x ! (n - x) !
%
n
n x(x - 1) n ! x n − x xn ! p q + ∑ x ! (n - x) ! ∑ x ! (n - x) ! p x q n − x x =0 x −0 n
∑
x =2
n(n - 1) (n - 2) ! px q( n-2) -( x-2) + E (x) (x - 2) ![(n - 2) - (x - 2)] !
= n (n-1) p2
n
∑
x =2
(n - 2) ! px-2 q( n-2) -( x-2) + np (x - 2) ![(n - 2) - (x - 2)] !
= n (n-1) p2 (q + p) n − 2 + np = n2 p2 – np2 + np
σ 2x
= E (x2) – [ E (x)]
2
= n2 p2 – np2 + np – n2 p2 = np – np2 = np (1 – p) = npq
σ 2 = npq H
! 2 % H 2 ; 2 ) 2 ! 0 " * + ; % σ =
σ2 =
127
K
npq
: 4
7 10 0 ( % 4 5 6 3 " 9 6 F % O F 8
3 & = % ; %
:
3 & = ! 0 % 1 " x D 1 2 1 q = 2 P
3 ! 0 " * + & x H 7 8
=
n = 100
= σ 2x = npq ; %
1 1 = 100 = 25 2 2 =σ =
npq =
25 = 5
D 2 4 2 & ! K ! ) * + / ) F ( + 7 . L 8
/ * ' C
F 2 ( 7 9 (P ... / ( ) ( 4 " )
2 ! 0 " * + / ) E (( ) " ) + 128
! / * ' C D 4 ( %
: ) * + ( " .
: " + 3 = 9 6 Q ! 0 ) ( . ( D ) ! ) C
R
.( ) ! ( 1 !
R
.( 4 ! ! ) ! ( &
R
./ ! ( + ! %
R
( :
( ) F S 4 ( % 4 ! F K %
3 & @ ) ! F K %
T 2 6 2 & ! 2 ( 8 ! ( % 5 G
)
R
R
R
/ 2 2 ( 4 H * 1 ! ) 1 H
( 2 + 2 ! 2 ( 8 ( + F ! ... 2 1 8
:( ( " . " , K ( 8
+ ! ( ) R1 F 1 8
.+ 8
( + F 8 4 F . R2
.+ H ( F 7 H * K ) +
129
( F 8 4 ( + F . " R3 .3 6 C L L 1 8
13 0
F 1 8
+
* 2 + ! 2 ! 0 % ) 1 ! * +
:! % K ) ! % )
X ! "
X
' ( ) X % & ... # 2 # 1 # 0
: * ' + .
f(x ) =
e−λ λx x!
... # 2 # 1 # 0 =
2. 7 18 28 . / =
x
,
e
+ " 4 0 1 & 2 3 ) ) ( + = λ . 5 ' 0 ) )
! " # $% # & ' ( # ) # *
:5 (
/ # # + , % - # " .
. / ' " % # $ % 0 #
13 1
:
! H 0 + ! 0 % 1 " x D / 2 ) . 3 ; ) λ ( ) ! ) * + f (x) =
e−λ λx x!
x 2 ( 4
: ) * + ( 1 8
e−3 (3) 4 4! 0.05 (81) = 24
p(x=4 ) = f (4 ) =
=
135 800
(e-3 =
0 .0 5 C )
= 0.16875
:6
F ( + ( ! K % ) H 6
: / 4 10 × 3 ( * / 4 20 .K 9 ;
. 4 G 9 K 9 ;
13 2
(i
(i i
:
10 × 3 ( ! K % " x D
! 3 % ! ) * + x 2 λ =
1. 5
10 × 3 ( ! K % ) ; e−λ λx f (x) = x!
i) P (x=0) = f (0) = 0. 223
3 × 10 = 1.5 20
: ) ( 1 8
,
e−1.5 (1.5)0 = e-1. 5 = 0!
ii) p ( 4 G 9 K ) = 1-p (x=0) = 1 - 0. 223
= 0. 7 7 7
/
) * + & ! ! ) 1 " x H 6
e−λ λx f (x) = x!
x = 0, 1, 2, … λ >
0
; * + ( % ; ) x ! 0 % ) 1 * 4
: ; λ ; )
13 3
µ x = E (x) = λ σ 2x = E (x ) – [ E (x)] = λ 2
E (x) =
2
∑x ∞
f (x)
x =0
∞
=
∑
=
∑
0
∞ 1
: 2 2
x
e-λ λx x!
e - λ λx ( x − 1) ! ∞
= λ e-λ ∑ 1
λx -1 ( x − 1) !
' ! %
eλ = 1 +
1 λx λ λ2 + + ... = ∑ 1! 2! x! x =0
∴ E (x) = λ e-λ eλ = λ E (x2) = =
∑ ∞
x =0
∞
∑ 0
x 2f ( x )
x 2 e-λ λx x!
W X(X-1)+ X ( X2 $
13 4
%
= = =
∞
∑ 0
[x (x - 1) + x] e-λ λx x!
∞ [x (x - 1) e - λ λx
∑
x!
0
∞
∑ 2
∞
+∑ 0
x e-λ λx + E (x) ( x − 2) ! ∞
= λ2 e −λ ∑ 2
λx - 2 +λ ( x − 2) !
= λ 2 e- λ e λ + λ = λ2 + λ v a r (x) = E (x2) – [ E (x) ] 2 = (λ2 + λ) - λ2 =λ
135
x e - λ λx x!
- 8 4 ) E ( 5 & 0 & L 8 20 3 , 8 (27/64)
: K ) F X * ( 0 (
(1
( ) K % 8 9 8 (i
(175/256) ( ) K % 4 G 9 8 9 8 (ii K K ( 0 ) 10 H 6 (2 :( 0. 2 ! 0 Z ) ; 9
(0. 1074)
(0. 8926)
Z ) ; 9 K K (i " Z ) 9 K
(0. 2)
(ii
( 0 ) 3 9 K (iii
% 3 + T 6 Q 6 0. 15 H 6 (3 ( 2 + & H ? . 5 . % 9 6 + & 15 E : ( )
K % + ; 3 (0. 0873)
(0. 3185)
(i
% 4 (ii
136
& = " k 4 H 6 (4
. k 2 ! * + * G 7 H ! ( ) 3
; 2 ) 2 H ? 5 & 7 ) (5 : E 1 /2 ; )
(1/16)
I I G *
(i
F 4 G 9 F ) G T (ii
(15/16)
! 2 % ! 4 2 L ! % & " " H (
: K ) J ( % 7 H H 7 + H ? E L '
(128/2187)
(448/2187) (1/2187)
(6
5 7 ; 9 8 (i
5 7 7 9 8 (ii
5 ' 7 ; 9 8 (iii
1 4 ) ! H 6 (7 ( 2 ) 7 2 ( ( ) ( ! ( ( !
2 2 4 @ H ! ( 0 / 4 ( % ) 7 ( " (0. 075)
(0. 225)
(0. 700)
:
( 1 M ( & *
(i
1 (ii
( 1 ( & " " (iii 137
2 & K ) 6 I ( % 4 & 9 8 (8 . 8 % I 9 L. 8 (63/256)
1/5 ( ! ( ) 7 K . & H 6 (9 .3 25 4 G 9 4 9 8 (0.766)
3 Z ) 9 K ( ! (10
2 I 2 Z ) 1/3 ( 8 .(16/243) ( 8
( 6 3 K Z )
6 ( 0 3 E ( 0. 30 $
(0. 58) O 4 G 9 /& " E A
(11
. 0. 90 ! 2 2 2 2 + + ! 2 ( 2 2 " ( 2 2 H 6 (12 D 4 E + + 10 3 2
(0. 6513)
3 9 3 ) ; H ! 0 " * + p , n % / 4 (13 ( n = 15 E p =
138
3/5) 18/5
E
p =
3 ; H ! 0 " * + * 4 (14 0. 2
(µ = 7
σ2 = 5 . 6 )
3 + ! % J 232 3 E ( 8 : ( 8
e-1/2 = )
(2e-1 = 0. 7 35 6 )
n = 35
232 Z K (15
K ) 0
J ' &
(i
J 4 &
(ii
(0. 18 39
C 4 * 8 ! % C ) H 6 (16 (e-4 = 0. 018 )
(0. 426)
: &
% & ! C ; *
(i
% & ! " G 9 C ( " . " * (ii
" ( + ! % ) H 6 (17 E ) * + * % H 6 , 0 4 I
:
139
(e-2 = 0. 135 ) , 0 4 I . * ; 8
(0. 382) , 0 4 10 . 8 % 4 "
(i
(ii
. ( * ) 9 (18 ) 8 ! ( & ) H 6 4
(0. 09)
O 4 !
T 6 & ! ( % 5 G ) H 6 (19 : E ( 8 ! 5 4 ) 5 3 8
(e-4=0. 018 ) (0. 91)
*
8 ( ! 4G 9 J K
(i
(ii
7 ( + 3 ! K % ) H 6 (20 R:/ 4 10 × 6 3 ( % / 4 F
(e-6=0. 0025 )
(0. 997 5 )
K )
K 4 G 9 & )
(i
(ii
( ( G ! [ C ) 3 Q X * 8 ! (21 ! K ) 200 [ ) L 0 . 500 3 " G 9
(0. 5 4 3)
(i
[ C . " 4 G 9 (ii
(0. 4 5 7 )
14 0
% / K 8 , 0 4 3 1 ! K C (22
. /7 × /3 % 3 4 E 2/2 ! (0. 00001)
K % 3
(0. 0001266)
(i
K " G 9 & (ii
14 1
@
( ) ( - 8 9 x 1 D (1 . 4 E 3 E 2 E 1 E 0 ! ( x / 4
p=
5 1 = H 0 8 K ) " p D 20 4
F X / K ) @ H ! 0 " x 1
9 6 1 8 K )
1 ; ) " 4
n= 4
(i
x= 1
1 4 3 q= 4
p=
: 9 8 ! 0 " * + ( 1 8
4 1 3 P (x =1) = f (1) = 1 4 4 1
3
4! 1 3 = ! ! 1 3 4 4 1 27 = 4
4 64 27 = 64
14 2
3
/ )
(ii P (x ≥ 1 ) = p (x =1) + p (x =2) + p (x = 3) + p (x = 4 ) = 1 – p (x =0)
4 1 3 = 1 – 0 4 4
0
3 = 1 - 4
4
4
= 1 – 8 1/ 25 6 = 17 5 / 25 6 x / 4 K & K ! ( 0 ) G " x D (2
0 E 1 E … E 10
! ( H 2
n = 10 p = 0. 2 q = 0. 8
i)
f (x) =
( ) px qn-x n x
p(x = 0) = f (0) =
! 0 " * + 3 1 8
( ) (0. 2)0 (0. 8 10 0
= (0. 8 )10 = 0. 107 4
14 3
)10
/ )
ii)
p(x ≥ 1) = f (1) + f(2) … +
f(10)
= 1 – f(0) = 1 – (0. 8 )10 = 0. 8 9 26 iii)
p (x = 3) = f(3) =
( ) (0.2) 10 3
3
(0.8) 7
= 120 (0. 008 ) (0. 8 )7 = 0. 2013 ≅ 0. 2
x / 4 . % 3 3 % F + & G " xD ( 3
0 E 1 E 2 E ... E 15 ! (
% + &
p = 0. 15
) + &
i) p (x = 0) = f(0) =
( ) (0. 15 15 0
q = 0. 85
n = 15
)0 (0. 8 5 )15
= (0. 8 5 )15 = 0. 08 7 3 ii) p ( x ≤ 1) = p (x = 0) + p (x =1) = f (0) + f(1) 14 4
=
( ) (0. 15 15 0
( )
1 14 )0 (0. 8 5 )15 + 15 1 (0. 15 ) (0. 8 5 )
= 0. 08 7 3 + 15 (0. 15 ) (0. 8 5 )14 = 0. 08 7 3 + 0. 2312 = 0. 318 5
: ! k 2 ( / (4 0E 1E 2E 3E 4
3 ) 3 9 8
n= 4
p=
q=
1 6 5 6
4 1 5 p (K = k) = f (k) = K 6 6 k
4−k
K 1 ! * + ! % !
4 K
Pk
Pk
0
(5 / 6)4
0. 48 23
1
4(1/ 6) (5 / 6)3
0. 38 5 8
2
6(1/ 6)2 (5 / 6)2
0. 115 7
3
4(1/ 6)3 (5 / 6)
0. 015 4
4
(1/ 6)4
0. 0008
145
! ( x / 4 G G " x D (5 0 E 1 E ... E 5
i)
p=
n= 5 1
2
q = 12
4G 9 7 7 ) G T
; E ( % 3 ". " " & . 5 5 &
; ) K
P = P(1) + P(2) + P(3) + P(4) =
=
( ) (1 2 )(1 2 ) + ( ) (1 2 ) (1 2 ) + ( ) (1 2 ) (1 2 ) + ( ) (1 2 ) (1 2 ) 4
5 1
5 2
2
3
5 10 10 5 15 + + + = 32 32 32 32 16
5 3
3
2
5 4
1
4
( ) + ( ) ) R1 = K (
= 1 - [P (5) + P (0)]
1 1 15 = 1- + =
ii)
32
32 16
/ & + / & ; ) K 146
P = P (0) + P (5 ) =
1 1 1 + = 32 32 16
7 D + & 8 ! 5 " x D (6 7 .... E 2 E 1 E 0 ! ( x / 4 E H 5 7 9 8 p D
5 7 9 8 / q p=
7 1 2 i) p (0) = f (0) = 0 3 3 0
q=
1 3 2 3
7
128 2 = = = 0.05853 3 2187 7
7 1 2 ii) p (1) = f (1) =
1 3 3
6
1 2 448 = 7 = = 0.2048 6
3 3
2187
7 1 2 iii) p (7 ) = f (7 ) = 7 3 3 7
0
1 1 = = = 0.0005 3 2187 7
E 0 ! ( x / 4 E 3 1 & " x D (7 9 ... E 2 E 1
147
n= 9
p= q=
1 4 3 4
3 1 M ( & * (i L 1 & 8
0 9 1 3 9 P (x = 0) = f (0) = 0 4 4 9 3 = 4
= 0. 07 5
1 (ii
9 1 3 P (x = 1) = f (1) = 1 4 4 1
8
= 0. 225
3 1 " " (iii P (x ≥ 2) = p (x = 2) + p (x = 3) + … + p (x = 9 ) = 1- [ p (x =0) + p (x = 1)] = 1 – 0. 07 5 – 0. 225 = 0. 7
148
! A
7 8 3 9 8 " x D (8
( x / 4 E 5 3 3 % 4 5
5 ... 2 E 0، 1 n= 5
8
p=q = 12
9 & 8 H 6 8 % I 9 . 8
; 5 4 3 2 1
(0 0) (1 1) (2 2) (3 3) (4 4) (5 5) ; ) 7 H ( 9 8 1 5 5 x 2
(0 , 0) = P(0) p(0)
()
0 5 5 1 1 = 0 2 2
()
5 1 = 50 2
L. "
()
1 0 1 5 5 0 2 2
2
8 % I 9 . 8 ∴
5 2 1 P = 50 + 2
()
2
()
()
()
1 5 1 5 5 5 5 + + 1 2 3 2 2 2
149
2
1 2
5 2
()
()
1 5 1 5 + 54 + 55 2 2 2
2
1 1 1 1 1 1 + 25 + 100 + 100 + 25 + 210 210 210 210 210 210
= =
252 210
=
252 63 = 1024 256
x / 4 E 3 25 3 G " x D (9 q =
4 5
E
25 ... E 1 E 0 ! ( p=
1 5
E
n = 25
4G 9 ( % 9 8
" ( % 9 8
P = (x = 4) + p (x = 5 ) + … + p (x = 25 ) = 1 – [ p (x = 0) + p (x = 1) + p (x = 2) + p (x =3)]
( )
( ) 15 45 + (252 ) 15
25 1 4 025 + 125 5 5 =1- 3 22 + 25 1 4 3 5 5
( )
24
2
4 5
23
= 1- (0. 0037 7 7 7 + 0. 0236110 + 0. 07 08 336 + 0. 135 7 644) = 0. 7 66
3 K Z ) I
Z ) 2 9 2 G ( % G ( 0) G 9 K ( 8 150
( 6
(10
( 8
( 6 I
Z ) 9 K ( 0
?
()
1 0 2 4 p = 04 3 3
:; )
()
1 1 2 0 11 3 3
2 1 16 16 = = = 3 3 35 243 4
P=
2 2 2 2 1 16 × × × × = 3 3 3 3 3 243
E 1 E 0 :! ( x / 4 " x D =
6 ...
p = 2 2 K
= q = 2 M K 6= n
2 4G 9 ( % " 6 5 4 3 2
P (x ≥ 2) = 1- p (x < 2)
= 1 – [ p (x = 0) + p(x = 1)]
H 2
151
(11
0. 30 0. 70
()
()
P = 1- [ 60 (0.3)0 (0.7)6 + 16 (0.3)1 (0.7)5 ]
= =
1 –( 0 .1 1 7 6 4 9 + 0 .5 7 9 8 2 5
0 .3 0 2 5 2 6 )
9 2 2 2 2 + + 2 H 6 * 2
/ 4 3 + + 1 " x D 2 4G
(12
! ( x
(0 E 1 E 2 E ... E 10)
0. 1 = p = D 4 =
0. 9 = q = D 4 / = 10 = n
= D 4 / A P = 1 –
4 = D 4
% & A
(100 ) ( 0 . 1 )
0
( 0 . 9 )10 = =
1 – ( 0 . 9 )10
4
0 .6 5 1 3
np ; ) ! 0 " * + * 4 ! ) )
npq ; ) ! 0 " * +
(13
n p = 9 … … … … … … … . (1 ) n p q =
18 … 5
… … … … …
(2 )
( 2) ( 1)
152
9 q =
18 5
q =
18 2 = 45 5
p =1-
2 3 = 5 5
n= 9 ÷
3 45 = = 15 5 3
(1 ) ! P ( 4 $
%
n = 35
(14
p = 0. 2
q = 0. 8
np = * 4
npq = µ = np ∴ µ = 35 (0. 2) = 7
σ 2 = npq
σ 2 = 7 (0. 8 ) = 5 . 6
* + x 2 E F ( 8 ! 5 G " x D (15 ( 8
! J ) )
λ =
232 =1 232
153
(i
e − λ λx x!
f ( x) =
p ( x= 2 ) =
e −1 (1) 2 2!
f(2 ) =
= e-1/ 2 =
0 .1 8 3 9
e = 0 .3 6 8 -1
P ( x<2 ) =
1 e −1 (1) x
∑
= =
(ii
x!
x =0
=
C 2
e-1 ( 1 +
1 ) 11
2 e-1
0 .7 3 6
) * + & % & ! C " x D (16 4 = 3 % ; H
e − 4 ( 4) 0
f ( 0 )= = P (x ≤ 3) =
(i
=
e
-4
0!
0 .0 1 8
∑
(ii
e - λ (λ) x x!
154
= p (x = 0) + p (x = 1) + p (x = 2) + p (x = 3) f(0) = e f(1) =
-4
= 0. 018
e−4 4 1!
= 0. 07 2
f(2) =
e − 4 (4) 2 = 0. 144 2!
f(3) =
e − 4 (4)3 = 0. 19 2 3!
f ( x ≤ 3) = 0. 426
% ! 1 ! x D (17 λ =
2 = ( 4 ! % ) 5
2 = , 0 4 5 ! % )
4 = , 0 4 10 ! % ) , 0 4 I !
i) P (x = 0) = f(0) =
e − 2 20 0!
=e
-2
= 0. 135
ii) λ = 4
, 0 4 !
P (x > 4) = 1-P (x ≤ 4)
155
λ = 2∴
λ = 4
e - λ λx P (x ≤ 4) = ∑ x! x =0 4
= P (x = 0) + P(x =1) + P(x =2) + P (x =3) + P(x =4) f(0) = e
-4
f(1) = e
-4
f(2) =
e − 4 (4) 2 =8 e 2!
f(3) = e f(4) =
(4)
-4
-4
64 6
e − 4 (4) 4 64 = e 4! 6
-4
=
0. 018
=
0. 07 2
=
0. 144
=
0. 19 2
=
0. 19 2
∴ P (x > 4) = 1 – P ( x ≤ 4 )
0. 618
= 1 – 0. 618 = 0. 38 2
4 . 8 ! ( & 1 " x D (18 λ = 2 3 % ) * + &
e − 2 24 P ( x = 4 ) = f (4) = 4
=
2 e 3
=
2(0.135) 270 = 3 3
-2
= 0. 09 0 156
( 8 ! ( 5 G 1 " x D (19 λ = 4 3 % ) * + x 2
i)
e − 2 24 P (x = 0) = f (0) = 0!
= e ii)
= 0. 018
-4
P(x ≥ 2) = 1 – P (x < 2) = 1 – [ P (X = 0) + (P(X =1)] = 1 – (0. 018 + 4 e
-4
)
= 1 – (0. 018 + 0. 07 2) = 0. 9 1
/ 4 10 × 6 3 ! K % " x D (20
6 =
(6) (10) = λ 3 % ; H ) * + x 2 10
f(x) = i)
e − λ λx x!
P (x = 0) = f(0) = e
0. 0025
157
-6
=
ii)
P (x ≥ 1) = 1- P (x < 1)
= 1-P (x = 0)
= 1 – f (0) = 1 - 0. 0025 = 0. 9 9 7 5
500 2 ! [ " x D (21 2 .5 =
500 = λ ∴ 200
f (x) =
i)
P (x≤2 ) =
f(0 ) +
f(0 ) = e
-2 . 5
f(1 ) = e
-2 . 5
f(2 ) = P ( x ≤2 ) =
ii)
f(1 ) +
f(2 )
e − 2.5 (2.5) 2 = 3.126 e - 2.5 2
6 .6 2 5 e
-2 . 5
=
( 6 .6 2 5 )( 0 .0 8 2 ) = 0 .5 4 3
1 – P ( x < 3)
P ( x ≥ 3) =
1 - {f (0) + f(1) + f(2)}
= =
1 –0 .5 4 3 =
e - λ λx x!
0 .4 5 7
158
/ 7 × 3 ( 4 ! K % " x D (22
f (x) =
e - λ λx x!
11.5 =
i)
P ( x= 0 )= f ( 0 )= e
ii)
P ( x ≤1 ) = f ( 0 ) + f ( 1 ) f(1 ) = e
-1 1 . 5
f ( 0 )+ f ( 1 )= 1 2 .5 e
-1 1 . 5
21 =λ 2
= 0 .0 0 0 0 1
1 1 .5 -1 1 . 5
159
= 0 .0 0 0 1 2 6 6
16 0
The Normal Distribution
2 " ( 2 % + / ! % * + %
2 ! 2 7 8 * ' 3 E , . 9 % )
H 2 J "( % " % + " 3 ) , 4 E 5 8
% + ) 3 % + / = % ? @ H E 3 4 .
/ 2 3 / , ) 5 % + ( +
2 2 " ! 2 * + H / ) E 3 4 . H J (7 ! 3 % ) ) 3 7 ( 8 K . M 3 % % + F % $
D e M o i v er ; / % * + H
H 2 % 1809 / ! G au s s I M / % 7 % 1733 /
H 2 & G au s s D i s t r i bu t i o n I 2 M * 2 + ; 3 ) L ' * + ( 2 8
( 2 2 2 ( ' 3 8 * + . ! 0 " * + 2 " T % +
2 / E ! % 9 9 * + H 9
2 1 2 ( 2 " ( 2 9 8 J ; = * + 9
" ! % * + 9 E * + H 3 2 ; H ! 0 %
2 . 2 2 " K ) ; ) ; H ! ) ) ! )
9 6 7 7 3 4 3 ! ) 4 . ) 16 1
2 & ! 2 G 7 K ) ) ( &
- 8 ; ) 9 ( ) @ H * (3 % 2 1 ; 2 ( 2 " 3 9 ( ) ! : 9 " 1
. 8 ! 0
1
2 2 & ( % ! 0 & @ µ (* 2 4 ) ! 2 ) 2 ) 2 2 ( 4 K)
$
% & ' %
2 ! 2 ( 2 % , 4 E σ ; %
4 E 2 ! ! ) ) ! & ; % 3 ! ; % ! ) ) ,
16 2
µ= 2
µ= 4
x
µ= 6
2
f(x) σ = 0 .5 σ =1 σ=2
x
µ= 3
3
9 ; µ ! ) ) " ! % 9 (1 .3 9 9 " ! ) ) ) 9
9 6 9 3 ) / ) 8 ! ! ) ) (2 . ) ) 4
:( A
. ; ) 9 3 ) D (3
! % 9 ? σ µ ( 4 & (4
µ + σ µ R σ * 3 ) % 68. 26 !
16 3
R
µ R2 σ * 3 ) % 95. 46 ! RK µ+ 2σ
3 σ µ R 3 σ * ( ) % 99. 74 ! RQ µ+
1. 96 σ µ R1. 96 σ * ( ) % 95 ! R µ+
2. 58 σ µ R2. 58 σ * ( ) % 99 ! R2 µ+
:! 4 ! - '
%68.26
13.6%
34 .13%
34 .13%
13.6%
2.14%
2.1 4 %
x
668.26 8 .26
95.46% 9 9 .7 4%
3 H 6 % D + 3 x 8 ! 0 % 1
:( ( 1 8 7 % 3 ( "
16 4
f (x) =
1
σ 2π
e
−
1 x - µ 2 2 σ
C
−∞ > x < ∞
! % 1 ! ) ) ! µ
2. 71828 3 4 " e
σ2 3 2
3. 14159 ; ) ( ( ) ! π
µ ! 2 ) 3 ) ; H ! % * + K
. " 2 N( µ , σ2 ) + K ( σ ; ) ; % 3 )
20 ; ) (* 4 ) ! ) ) 3 % 1 ! % N (2 0,4 ) .(2 ; ) % ; ) 4 ; )
2 ( 2 ) D 2 ( " 3 f (x)
. 8 0 L 1 - 8
) 3 7 H 9
< x < b ) b , a " 4 * σ2 3 µ 3 ) x=
f (x) 3 9 7 8 ( ) ; ) p (a
b,x= a
f(x) µ
16 5
a
b
5
The Standard Normal
D i s tri b u ti on
1 9 ! ) ; % ! % 1 / ) , 1 9 ; 3 8
! ) 3 ) ; H ! %
! 3 ( " 3 z + 3 + N(0.1) 1 e- z 2π
f (x) =
2 /2
−∞ < z < ∞
f (z) 9 ( ) D ( " 3 f (z)
∫
∞ -∞
1; )
1 -z 2 / 2 e d z= 1 2π
z z ; % ! % ! 0 % 1 D 4
f 2 ( " 3 9 7 8 ( ) z = b E = a
; ) ; z = b E z = a (z)
P (a < z < b ) =
b
∫
a
1 2π
2 e− z / 2 d z
! ; % ! % 9 ( ) ( ) ) ., 2 ! 7 %
4 8 ! % 1 D 4 K )
f (x) 2 ( 2 " 3 9 7 8 ( ) K) ) % ' f(x) 16 6
X 2
7 M 4 H J & σ2 µ ! )
4 L ' H J 3 ! 8 1 ! 0 % 1 6 3 % 4 ) K ) M 3 ? T ; 7 M
* 2 ) x , σ2 , µ 3 / ! % 9
9 2 6 N ( µ , σ2 ) ! 2 % 1 ( % 8 7 H 9 K 1 ! 7 3 ) K ) N(0.1) ; % 1
., 2 ! ) 7 %
N ( µ , σ2 )
N( 0 . 1 )
? N ( µ , σ2 ) ! % * + 3 x ! 0 % 1 H 6
; 2 % ! 2 % * 2 + 3 2 z= 3 8
x−µ σ
! 0 % 1
! ) 3 ) % % L % + 3 ; N(0.1)
9 6 3 σ2 3 µ ! ) 3 ) ! % 1 ; 1 ! 2 % * + H & z=
x−µ 3 ; % ! % σ
1
7 7 8 ( ) K ) ; % ! % * + 9 6 .6 !
P (x ≤ b) =
x / 4 ( % ( 4 b H 6
P (x - µ ≤ b - µ ) 167
x−µ b−µ P ≤
= =
σ
b−µ P z ≤
σ
σ
f(x)
µ
168
b
x
f(z)
Q
z =
6
z
b−µ σ
z N ( µ , σ 2 )
P (X ≤ b) = P (Z ≤
! 9 3 = M ( )
b-µ ) σ
P (X > b) = 1 – P (X ≤ b)
'
b-µ ) σ
= 1 – P (Z ≤
(7 ! ) a< b H 6
P (a ≤ X ≤ b) = P (Z ≤ a ) 9 6 ( ) A
169
b-µ a -µ )- P (Z ≤ ) σ σ
4 b ) 9 6 ( ) ; ) ;
x
a b
µ
7
8 ( " 3 9 " ( 8
i)
x
∫
−∞
f(z)d z = =
ii)
0
∫
−∞
1
2
f(z) d z +
∫ f(z) d z
x o
∫ f(z) d z
x
+
o
−x
∞
−∞
x x
∫ f(z) d z = ∫ f(z) d z =
1
2
-
∫ f(z) d z
o
2 ) / ( ) ! ! @ H
3 2 x / ( K ) ; % ! % 9 .( ) x / Q ) &
170
: :1
2 N (3 E 4 ) * + 3 ! % ! 0 1 X H 6
O5 E 3 X *
:
x−µ σ 3−3 z1 = =0 2 5−3 z2 = =1 2
z=
; % * + 9 6 * +
p (3 < x < 5) = p (0 < z < 1)
! ! !
z= 1 z= 0 3 ) ; )
0
1
0. 3413 ; ) , 2 !
171
:2
N ( µ,σ2 ) ! % * + 3
X ! 0 % 1 H 6
i)
P ( µ − σ < x < µ + σ)
ii)
P ( µ − 2σ < x < µ + 2σ )
iii) P ( µ − 3σ < x < µ + 3σ) i)
z1 = z2 =
:
:
(µ − σ ) − µ = −1 σ (µ + σ) − µ =1 σ
∴ P ( µ − σ < x < µ + σ) = P( −1 < z < 1)
z=1
z=1
z = -1 ( ) F
z=0 ( ) + z=0
-1
0
1
µ=0 ! ) ) ) 9
z=1
z=1 ( ) =
Z 9 " (
z = 0 ( ) ; ) z = 0 z = -1 ( )
0. 3413 ; ) 7 . ! 172
∴ P ( µ − σ < x < µ + σ ) = 0. 34 13 +
0. 34 13
= 0. 6 8 26
'
= 6 8 . 26 %
* 9 ( ) D % 68. 26 ! % H
.! ) ) ; %
ii)
P ( µ − 2σ < x < µ + 2σ ) (µ − 2σ) − µ = −2 σ (µ + 2 σ ) − µ z2 = =2 σ
z1 =
∴ P ( µ − 2σ < x < µ + 2σ) = P( −2 < z < 2)
F 8 ( ) + z = 0 z = -2 7 8 ( ) ; )
z =2 z =0
= 0. 4 7 7 3 + 0. 4 7 7 3 = 0. 9 5 4 6
'
=9 5 .4 6 %
* 9 ( ) D % 95. 46 ; . 8
! ) ) %
Z 173
-2
iii)
2
(µ − 3σ) − µ = −3 σ (µ − 3σ) − µ =3 z1 = σ
z1 =
∴ P ( µ − 3σ < x < µ + 3σ) = P(−3 < z < 3)
F 8 ( ) + z = 0 z = -3 7 8 ( ) ; )
: z = 3 z =0
= 0. 4 9 8 7 + 0. 4 9 8 7 = 0. 9 9 7 4 =9 9 .7 4 %
3 2 '
* 2 9 2 2 ( ) D % 99. 74 ; . ! ) ) 3 %
99.74%
-3 3
174
:( ; % ! % * + .1
(0. 2580)
P (0 < z < 0. 7)
(0. 3849)
P (0 < z < 1. 2)
(0. 8643)
(ii
P (R1 < z < 0)
(iii
P (R0. 82 < z < 0. 96)
(iv
(0. 3413)
(0. 2742)
(i
P(
P(
(0. 6254)
z < 0. 6R)
z > R1. 1)
(v
(vi
z = R0. 5 ( ) .2
: N (2, 1) ! % * + 3 x ! 0 % 1 H 6 .3 (0. 0227)
P (x > 4 )
(0. 4773)
P (0 < x < 2 )
(i
(ii
2 & 2 % 3 2 + % T K. H 6 .4
3 2 3 8 68. 50 ! ) 3 ) ; H ! % * + :3 8 2. 3 ; %
175
+ ( % K ;
(i
/ & H ( % ! 3 ( ) !
(ii
.(0. 0643) (3 8 72) / 4 6
(0. 1935) 3 8 72 70
2 ! % * + H K 3000 H 6 .5
2 / 2 ) 170 ; 2 ) 2 G 7 H & ! ) ) : / ) 5 ; ) & ; %
./ ) 185 " / & H ( ( ) ./ ) 185 / & + H (
2 ) , 2 / & H ( ( ) H 6 2 0. 2881 x % 4
(i
(ii
(iii
.O H
3 2 ) ; H 2 ! 2 % * + 3 x ! 0 % 1 H 6 .6 : J 0. 30 ; % 3 80 ! ) P (X ≤ 80. 36)
P ( X ≤ C) = 0. 95 H 6 C % 1
4
& 4 $
(i
(ii
G W * 8 .7
* + & * 8 H Q G 4 7 / )
E / 2 0. 0025 ; % 3 / 2. 5 ! ) 3 ) ! % 176
2 2 4 2 ! 2 a ( 0 ( ) !
O/ 2. 5049 2. 4951
; % ( ) ! 4 % ) H 6 .8 ( 2 ) ! % % + * G H 6 0. 50 O( ) ! 3 2. 5 ' H %
2 ( ) 1500 ! 0 & 8 ) H 6 .9 : % % + * ( ) 50 ; % .( ) 1400 4 , ) 8
(i
.( ) 1550 " S % ) 8 6
(ii
.( ) 1550 1450 S % ) 8 6 (iii I 2 10 ; ) 3 3 + , 8
.10
3 + + I 1. 5 ; % (O U NCE S )
12. 4 7. 9 2 & + * ! ( ) ! %
OI
* 2 + 2 & ( ) ! T 6 H 6
5 ; 2 % 3 2 100 3 ) ; H ! % : .
. 100 + ) (
. 100 ) ( 177
.11 (i
(ii
110 100 2 ) ( (iii .
178
( )
.7 , 2 ( .1 ! 2 2 ( 2 2 ( 2 2 ) .2
2 ! 2 ( 2
= ( )
R0. 5 R ∞ 2 F 8 0.5 2 ( 2 ) " ! 2 ( 2 ) A
4 ∞ 9
= 0. 5 9 8
0. 5 – 0. 1915 = 0. 3085
; % * + 9 6 ! % * + .3 z= z=
z -µ σ
4-2 =2 1
∴ P (X > 4 ) = P (z > 2)
0
= 0. 5 – 0. 4 7 7 3
2
Z Z1 =
179
= 0. 0227 -0 . 5
0 - 2 = -2 1
(ii
(i
∴
Z2 =
P (0 < X < 2)
2 - 2 =0 1
= P (z > 2) = P (-2 < z < 0) = P (0 < z <2)
=
0. 4 7 7 3
.4 x -µ σ 72 - 68.5 3.5 = = 2.3 2.5
.i
Z =
= 1. 5 2
∴ p (x > 7 2) = p (z > 1. 5 2)
0
= 0. 5 -0. 4 35 7 = 06 4 3 70 - 68.5 = 0. 6 5 2.3 72 - 68.5 z2 = = 1. 5 2 2.3
.ii
z1 =
∴ P (7 0 < x < 7 2) = P (0. 6 5 <
z < 1. 5 2) 18 0
= 0. 4 35 7 – 0. 24 22 = 0. 19 35
185 − 170 =3 5
z=
. i .5
∴ P (x > 18 5 ) = P ( z > 3 )
= 0. 5 – 0. 4 9 8 7 = 0. 0013
3
0
3000 = %
100
x = %
0. 13
(0.0013) (3000) 0.100
x=
= 3. 9 ≅ 4 z =
0 . 28 8 1
.ii
x
- 170 5
7 = z ( 4
.iii
x
0.8 ! 0. 2881 ( )
∴ 0. 8 =
0
x = 18 1
χ - 170 5
174
Z =
.i .6
80 - 36 - 80 = 1.2 0 .3
∴ p ( x ≤ 80.36 ) = p ( z ≤ 1.2)
= 0. 5 +
1 .2
0. 38 49
= 0. 8 8 49
0 .4 5
z=
.ii
c − 80 0 .3
z ( 2 4 2
0
– 0. 50) ( 2 2 ) ( 2 2 0
! 0. 45 ( ) ; (0. 95
144424443
1. 64
0.95 =
0.4 5
∞ − ∞ ! " ∞
∴
c − 80 = 1.64 0 .3
∴ c = 8 0. 49 2
z1 = z2 =
2.4951 − 2.5 = 1.96 0.0025 2.5049 − 2.5 = 1.96 0.0025
.7
∴ P (2. 49 51 < x < 2. 5049 ) = P
(-1. 9 6 < z < 1. 9 6 )
-1. 9 6 1. 9 6
= 2 (0. 4750)
182
= 0. 9 5 z1 = z2 =
-3 2
.8
2 .5 − 4 =-3 0 .5 3−4 =-2 0 .5
∴ p (2. 5 < x <3) = p (-3 < z <-
-
2)
= p (2 < z < 3) = 0. 49 8 7 – 0. 4773 = 0. 0214
z=
1400 - 1500 = -2 50
.i .9
∴ p (x < 1400) = p (z < -2 )
-2
= 0. 5 – 0. 4773
0
= 0. 0227 z=
1550 - 1500 = 1 50
.ii
∴ p ( x > 1550) = p (z > 1 )
0
= 0. 5 – 0. 3413
1 183
= 0. 158 7
-1 1
z1 =
1450 - 1500 = -1 50
z2 =
1550 - 1500 50
.iii
=1
∴ p (1450 < x < 1550) = p (-1
< z <1 ) 0
= 2 (0. 3413) = 0. 6 8 26 7.9 - 10 = 1.4 1.5 12.4 - 10 z2 = = 1.6 1.5
.10
z1 =
-1. 4
0
∴ p (7. 9 < x < 12. 4 ) = p (-1. 4
< z < 1. 6 )
1. 6
= 0. 419 2 + 0. 4452 = 0. 8 6 44
.11 .i
184
z=
100 - 100 =0 5
p ( x > 100 ) = p (z > 0 ) = 0. 5
z=
.ii
100 - 100 =0 5
p ( x < 100 ) = p (z > 0 ) = 0. 5
.iii
100 - 100 =0 5 100 - 100 =2 z2 = 5
z1 =
∴ p (100< x < 110) = p (0 < z <
2)
= 0. 4773
185
2 ) 2 3 & ( 0 ! ( 0 % ( R1 2 K ) .F ) ) I K ) G 5 Z
. ( 0 % (3 /8 )
= 0. 5 2
P(A2) = 0. 7
S ! 2 % c ! " A2 E A1 H 6 R 2
P(A1)
: P(A1 A2) = 0. 3
" A2 A1
(0. 9)
( ; ) & D ) P ( A 1 A 2)
(0. 2)
P ( A 2 R A 1)
(i
(ii
(iii
P ( A1 R A2) (iv
( ) )
= 0. 4 2 S ! 2 % c ! " P(B) = 0. 7
(0. 8)
B
A H 6 R3 P(A)
: K ) P(AB) = 0. 3
B A " D 4
186
(i
& I B A D 4
(ii
AD 4 /
(iii
(0. 5)
(0. 6)
(0. 1)
BD 4 / AD 4
(iv
: ( C 7 7 ! R4
(6/36)
7 = 4 D A1
(i
10 " = 4 D A2 (ii (3/36)
5 ; 2 ) 2 2 4 2 = 4 D A3 (iii (10/36)
4 G 9 7 7 & = 2 % A4 (iv (11/36)
(3 /6 )
: ( K ) , ) I ! R5
6 G $
! + D $
9 ! + D
9 6 ! "
(i
(ii
(3/18)
(4/6) 5 4 D H 6 ! + D (iii (1 /6 )
4 G $
9 6 ! "
(9/18) ; G H 6 ; D
187
(iv (v
( 8 H + M 9 6 % & 4 8
. + 2 1 ( 8 6 K ) J
3 R6
1 ! + 1 ( 8 6 ! & 3
(5 /9 )
2 % 50 & % % 20 P ) A H 6 R7 2 % 90 2 ! 4 B P ) 5 &
2 & % 2 K K ) H ? .5 &
.5 K K )
(0. 82)
5 2 3 K ) K H 6 , ) Z ) ! R8 O 3 ! ! A
(10/18=0. 5555)
/ 0. 40 = 5 8 F ! K H 6 R9 (0. 2592)
(0. 92224)
(0. 98976)
(0. 07776)
:
K . 5 K -
(i
4G 9 K -
(ii
-
(iv
" G 9 K . 4 - (iii
! C . " 5 D 9 8 R10 O 6 ! +
(0. 17342)
188
2 0. 3 & [ 8 ( 8 6 H 6 R11
9 & ( 8 6 ! & 4. 6 / + . P 8 .% 80 4 G
0. 7 = J
(0. 7)n = d P 8 1-(0. 7)n > 0. 8 K
(0. 7)n > R0. 2 (0. 7)n < 0. 2
(0. 7)1 = 0. 7 , (0. 7)2 = 0. 49 (0. 7)3 = 0. 743 , (0. 7)4 = 0. 2401 (0. 7)5 = 0. 16 8 07
∴ n = 5 P 8
189
:
19 0
(1)
e-x x
e-x
x
e-x
x
0.0
1 .000
2 .0
0.1 35
4.0
0.1
0.9 05
2 .1
0.1 2 2
0.2
0.8 1 9
2 .2
0.3
0.7 41
0.4
e-x
x
e-x
x
0.01 8
6.0
0.002 5
8 .0
0.00034
4.1
0.01 7
6.1
0.002 2
8 .1
0.00030
0.1 1 1
4.2
0.01 5
6.2
0.002 0
8 .2
0.0002 8
2 .3
0.1 00
4.3
0.01 4
6.3
0.001 8
8 .3
0.0002 5
0.67 0
2 .4
0.09 1
4.4
0.01 2
6.4
0.001 7
8 .4
0.0002 3
0.5
0.607
2 .5
0.08 2
4.5
0.01 1
6.5
0.001 5
8 .5
0.0002 0
0.6
0.549
2 .6
0.07 4
4.6
0.01 0
6.6
0.001 4
8 .6
0.0001 8
0.7
0.49 7
2 .7
0.067
4.7
0.009
6.7
0.001 2
8 .7
0.0001 7
0.8
0.449
2 .8
0.061
4.8
0.008
6.8
0.001 1
8 .8
0.0001 5
0.9
0.407
2 .9
0.055
4.9
0.007
6.9
0.001 0
8 .9
0.0001 4
1 .0
0.368
3.0
0.050
5.0
0.0067
7 .0
0.0009
9 .0
0.0001 2
1 .1
0.333
3.1
0.045
5.1
0.0061
7 .1
0.0008
9 .1
0.0001 1
1 .2
0.301
3.2
0.041
5.2
0.0055
7 .2
0.0007
9 .2
0.0001 0
1 .3
0.2 7 3
3.3
0.037
5.3
0.0050
7 .3
0.0007
9 .3
0.00009
1 .4
0.2 47
3.4
0.033
5.4
0.0045
7 .4
0.0006
9 .4
0.00008
1 .5
0.2 2 3
3.5
0.030
5.5
0.0041
7 .5
0.00055
9 .5
0.00008
1 .6
0.2 02
3.6
0.02 7
5.6
0.0037
7 .6
0.00050
9 .6
0.00007
1 .7
0.1 8 3
3.7
0.02 5
5.7
0.0033
7 .7
0.00045
9 .7
0.00006
1 .8
0.1 65
3.8
0.02 2
5.8
0.0030
7 .8
0.00041
9 .8
0.00006
1 .9
0.1 50
3.9
0.02 0
5.9
0.002 7
7 .9
0.00037
9 .9
0.00005
19 1
e-x
! % 9 ) /
(2 ) / 2 4 , 2
( 2,0 A)
0 Z 0.00 .01 .02 .03 .04
A 0.0000 .004 0 .008 0 .01 20 .01 6 0
Z 0.4 7 .4 8 .4 9 .5 0 .5 1
A 0.1 8 08 .1 8 4 4 .1 8 7 9 .1 9 1 5 .1 9 5 0
Z 0.9 4 .9 5 .9 6 .9 7 .9 8
A 0.326 4 .328 9 .331 5 .334 0 .336 5
.05 .06 .07 .08 .09
.01 9 9 .0239 .027 9 .031 9 .035 9
.5 .5 .5 .5 .5
2 3
.1 9 8 5 .201 9 .205 4 .208 8 .21 23
.9 9 1 .00 1 .01 1 .02 1 .03
.1 .1 .1 .1 .1
0
.039 8 .04 38 .04 7 8 .05 1 7 .05 5 7
.5 .5 .5 .6 .6
7
.21 5 7 .21 9 0 .2224 .225 8 .229 1
.1 .1 .1 .1 .1
5
9 6 36 5 4
.6 .6 .6 .6 .6
2 3
9
.05 .06 .06 .07 .07
.20 .21 .22 .23 .24
.07 .08 .08 .09 .09
9 3 32 7 1 1 0 4 8
.6 .6 .6 .7 .7
7
.25 .26 .27 .28 .29
.09 8 7 .1 026 .1 06 4 .1 1 03 .1 1 4 1
.7 .7 .7 .7 .7
2 3
30 .31 .32 .33
.1 .1 .1 .1
.7 .7 .7 .8
1
2 3 4 6 7 8
7 5
1 7 21 25 29
1 4
5
7
9 3
4 5 6 8 9
0 1
4 5 6 8 9
0 1
4 5
9
6 8
7 0
z
Z 1 .4 1 1 .4 2 1 .4 3 1 .4 4 1 .4 5
A 0.4 207 .4 222 .4 236 .4 25 1 .4 26 5
.338 9 .34 1 3 .34 38 .34 6 1 .34 8 5
1 .4 1 .4 1 .4 1 .4 1 .5
6
.4 .4 .4 .4 .4
27 9 29 2 306 31 9 332
1 .04 1 .05 1 .06 1 .07 1 .08
.35 .35 .35 .35 .35
08 31
1 .5 1 .5 1 .5 1 .5 1 .5
1
.4 .4 .4 .4 .4
34 35 37 38 39
.2324 .235 7 .238 9 .24 22 .24 5 4
1 .09 1 .1 0 1 .1 1 1 .1 2 1 .1 3
.36 .36 .36 .36 .37
21
1 .5 1 .5 1 .5 1 .5 1 .6
6
.4 .4 .4 .4 .4
4 06 4 1 8 4 30 4 4 1 4 5 2
.24 .25 .25 .25 .26
8 6
.37 .37 .37 .37 .38
29 7 0 9 0 1 0
1 .6 1 .6 1 .6 1 .6 1 .6
1
8 0 1 2
1 .1 1 .1 1 .1 1 .1 1 .1
.4 .4 .4 .4 .4
4 6 3 4 7 4 4 8 5 4 9 5 5 05
.26 .26 .27 .27 .27
4 2 7 3 04 34 6 4
1 .1 9 1 .20 1 .21 1 .22 1 .23
.38 .38 .38 .38 .39
30 4 9 6 9 8 8 07
1 .6 1 .6 1 .6 1 .6 1 .7
6
.4 .4 .4 .4 .4
5 1 5 5 25 5 35 5 4 5 5 5 4
.27 28 .28 .28
9 4 23 5 2 8 1
1 .24 1 .25 1 .26 1 .27
.39 .39 .39 .39
25
1 .7 1 .7 1 .7 1 .7
1
1 8 4 9
19 2
4 5 6 7 8
5 4 7 7 9 9 4 3 6 5 8 6 08 4 9
4 4 6 2 8 0
7 8 9
0 2 3 6 5 7 8 9
0 2 3 4 5 7 8 9
0 2 3 4
5 7
0 2 4
.4 5 6 4 .4 5 7 3 .4 8 2 .4 5 9 1
.34
.1 331
.8 1
.29 1 0
1 .28
.39 9 7
1 .7 5
.4 5 9 9
.35 .36 .37 .38 .39
.1 .1 .1 .1 .1
36 8 4 06 4 4 3 4 8 0 5 1 7
.8 .8 .8 .8 .8
2 3 4 5 6
.29 39 .29 6 7 .29 9 6 .3023 .305 1
1 .29 1 .30 1 .31 1 .32 1 .1 33
.4 .4 .4 .4 .4
01 5 032 04 9 06 6 08 2
1 .7 1 .7 1 .7 1 .7 1 .8
4 .4 .4 .4 .4
6 08 6 1 6 6 25 6 33 6 4 1
.4 .4 .4 .4 .4 .4 .4
.1 .1 .1 .1 .1 .1 .1
5 5 4 5 9 1 6 28 6 6 4 7 00 7 36 7 7 2
.8 .8 .8 .9 .9 .9 .9
7
2 3
.307 9 .31 06 .31 33 .31 5 9 .31 8 6 .321 2 .3238
1 .34 1 .35 1 .36 1 .37 1 .38 1 .39 1 .4 0
.4 .4 .4 .4 .4 .4 .4
09 9 1 1 5 1 31 1 4 7 1 6 2 1 7 7 1 9 2
1 .8 1 .8 1 .8 1 .8 1 .8 1 .8 1 .8
.4 .4 .4 .4 .4 .4 .4
6 4 9 6 5 6 6 6 4 6 7 1 6 7 8 6 8 6 6 9 3
0 1
2 3 4 5 6
8 9
0 1
Z 1 .8 8 1 .8 9 1 .9 0
A 0.4 7 00 .4 7 06 .4 7 1 3
Z 2.4 1 2.4 2 2.4 3
A 0.4 9 20 .4 9 22 .4 9 25
Z 2.9 4 2.9 5 2.9 6
A 0.4 9 8 4 .4 9 8 4 .4 9 8 5
1 .9 1 .9 1 .9 1 .9 1 .9
.4 .4 .4 .4 .4
7 1 9 7 26 7 32 7 38 7 4 4
2.4 2.4 2.4 2.4 2.4
.4 .4 .4 .4 .4
9 27 9 29 9 31 9 32 9 34
2.9 7 2.9 8 2.9 9 3.00 3.1
.4 .4 .4 .4 .4
1 .9 6 1 .9 7 1 .9 8 1 .9 9 2.00
.4 .4 .4 .4 .4
7 5 0 7 5 6 7 6 2 7 6 7 7 7 3
2.4 2.5 2.5 2.5 2.5
9
.4 .4 .4 .4 .4
9 36 9 38 9 4 0 9 4 1 9 4 3
3.2 3.3 3.4 3.5 3.6
2.01 2.02 2.03 2.04 2.05
.4 .4 .4 .4 .4
7 7 8 7 8 3 7 8 8 7 9 3 7 9 8
2.5 2.5 2.5 2.5 2.5
4
.4 .4 .4 .4 .4
9 4 5 9 4 6 9 4 8 9 4 9 9 5 1
2.06 2.07 2.08 2.09 2.1 0
.4 .4 .4 .4 .4
8 03 8 08 8 1 2 8 1 7 8 21
2.5 2.6 2.6 2.6 2.6
9
.4 .4 .4 .4 .4
2.1 2.1 2.1 2.1
.4 .4 .4 .4
8 26 8 30 8 34 8 38
2.6 2.6 2.6 2.6
4
.4 .4 .4 .4
1 4
2 3
1
5
4
2 3
4 5 6 7 8 0 1
2 3 5 6 7 8
1
0
7
6
5
2 3
6 7 8 9
0 1
2 3 4 5 6 7
... Z 3.4 7 3.4 8 .4 9
A 0.4 9 9 7 .4 9 9 8 .4 9 9 8
9 8 5 9 8 6 9 8 6 9 8 7 9 8 7
3.5 3.5 3.5 3.5 3.5
0
.4 .4 .4 .4 .4
9 9 8 9 9 8 9 9 8 9 9 8 9 9 8
.4 .4 .4 .4 .4
9 8 7 9 8 8 9 8 8 9 8 9 9 8 9
3.5 3.5 3.5 3.5 3.5
5
.4 .4 .4 .4 .4
9 9 8 9 9 8 9 9 8 9 9 8 9 9 8
3.7 3.8 3.9 3.1 0 3.1 1
.4 .4 .4 .4 .4
9 8 9 9 9 0 9 9 0 9 9 0 9 9 1
3.6 3.6 3.6 3.6 3.6
0
.4 .4 .4 .4 .4
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
9 5 2 9 5 3 9 5 5 9 5 6 9 5 7
3.1 3.1 3.1 3.1 3.1
.4 .4 .4 .4 .4
9 9 1 9 9 1 9 9 2 9 9 2 9 9 2
3.6 3.6 3.6 3.6 3.6
5
.4 .4 .4 .4 .4
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
9 5 9 9 6 0 9 6 1 9 6 2
3.1 7 3.1 8 3.1 9 3.20
.4 .4 .4 .4
9 9 2 9 9 3 9 9 3 9 9 3
3.7 3.7 3.7 3.7
0
.4 .4 .4 .4
9 9 9 9 9 9 9 9 9 9 9 9
19 3
2 3 4 5 6
1
2 3 4 6 7 8 9 1
2 3 4 6 7 8
1
9
2 3
2.1 5
.4 8 4 2
2.6 8
.4 9 6 3
3.21
.4 9 9 3
3.7 4
.4 9 9 9
2.1 6 2.1 7 2.1 8 2.1 9 2.20
.4 .4 .4 .4 .4
8 4 6 8 5 0 8 5 4 8 5 7 8 6 1
2.6 2.7 2.7 2.7 2.7
9
.4 .4 .4 .4 .4
9 6 4 9 6 5 9 6 6 9 6 7 9 6 8
3.22 3.23 3.24 3.25 3.26
.4 .4 .4 .4 .4
9 9 4 9 9 4 9 9 4 9 9 4 9 9 4
3.7 3.7 3.7 3.7 3.7
5
.4 .4 .4 .4 .4
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
2.21 2.22 2.23 2.24 2.25
.4 .4 .4 .4 .4
8 6 5 8 6 8 8 7 1 8 7 5 8 7 8
2.7 2.7 2.7 2.7 2.7
4
.4 .4 .4 .4 .4
9 6 9 9 7 0 9 7 1 9 7 2 9 7 3
2.27 3.28 3.29 3.30 3.31
.4 .4 .4 .4 .4
9 9 5 9 9 5 9 9 5 9 9 5 9 9 5
3.8 3.8 3.8 3.8 3.8
0
.4 .4 .4 .4 .4
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
2.26 2.27 2.28 2.29 2.30
.4 .4 .4 .4 .4
8 8 1 8 8 4 8 8 7 8 9 0 8 9 3
2.7 2.8 2.8 2.8 2.8
9
.4 .4 .4 .4 .4
9 7 4 9 7 4 9 7 5 9 7 6 9 7 7
3.32 3.33 3.34 3.35 3.36
.4 .4 .4 .4 .4
9 9 6 9 9 6 9 9 6 9 9 6 9 9 6
3.8 3.8 3.8 3.8 3.8
5
.4 .4 .5 .5 .5
9 9 9 9 9 9 000 000 000
2.31 2.32 2.33 2.34 2.35
.4 .4 .4 .4 .4
8 9 6 8 9 8 9 01 9 04 9 06
2.8 2.8 2.8 2.8 2.8
.4 9 7 7 .4 9 7 8 .4 9 7 9 .4 9 8 0 .4 8 0
3.37 3.38 3.39 3.4 0 3.4 1
.4 .4 .4 .4 .4
9 9 6 9 9 6 9 9 7 9 9 7 9 9 7
2.36 2.37 2.38 2.39 2.4 0
.4 .4 .4 .4 .4
9 09 9 1 1 9 1 3 9 1 6 9 1 8
2.8 2.9 2.9 2.9 2.9
.4 .4 .4 .4 .4
3.4 3.4 3.4 3.4 3.4
.4 .4 .4 .4 .4
9 9 7 9 9 7 9 9 7 9 9 7 9 9 7
0 1
2 3 5 6 7 8 0 1
2 3 4 5 6 7 8 9 1
0 2 3
9 8 1 9 8 1 9 8 2 9 8 3 9 8 3
19 4
6
5
4
2 3
6 7 8
1
9
9
8
7
6
4
2 3
19 5
2 ) 2 3 & ( 0 ! ( 0 % ( R1 2 K ) .F ) ) I K ) G 5 Z
. ( 0 % (3 /8 )
= 0. 5 2
P(A2) = 0. 7
S ! 2 % c ! " A2 E A1 H 6 R 2
P(A1)
: P(A1 A2) = 0. 3
" A2 A1
(0. 9)
( ; ) & D ) P ( A 1 A 2)
(0. 2)
P ( A 2 R A 1)
(i
(ii
(iii
P ( A1 R A2) (iv
( ) )
= 0. 4 2 S ! 2 % c ! " P(B) = 0. 7
(0. 8)
B
A H 6 R3 P(A)
: K ) P(AB) = 0. 3
B A " D 4
19 6
(i
& I B A D 4
(ii
AD 4 /
(iii
(0. 5)
(0. 6)
(0. 1)
BD 4 / AD 4
(iv
: ( C 7 7 ! R4
(6/36)
7 = 4 D A1
(i
10 " = 4 D A2 (ii (3/36)
5 ; 2 ) 2 2 4 2 = 4 D A3 (iii (10/36)
4 G 9 7 7 & = 2 % A4 (iv (11/36)
(3 /6 )
: ( K ) , ) I ! R5
6 G $
! + D $
9 ! + D
9 6 ! "
(i
(ii
(3/18)
(4/6) 5 4 D H 6 ! + D (iii (1 /6 )
4 G $
9 6 ! "
(9/18) ; G H 6 ; D
19 7
(iv (v
( 8 H + M 9 6 % & 4 8
. + 2 1 ( 8 6 K ) J
3 R6
1 ! + 1 ( 8 6 ! & 3
(5 /9 )
2 % 50 & % % 20 P ) A H 6 R7 2 % 90 2 ! 4 B P ) 5 & 2 & % 2 K K ) H ? .5 &
.5 K K ) (0. 82)
5 2 3 K ) K H 6 , ) Z ) ! R8 O 3 ! ! A
(10/18=0. 5555)
/ 0. 40 = 5 8 F ! K H 6 R9 (0. 2592)
(0. 92224)
(0. 98976)
(0. 07776)
:
K . 5 K -
(i
4G 9 K -
(ii
-
(iv
" G 9 K . 4 - (iii
! C . " 5 D 9 8 R10 (0. 17342)
O 6 ! +
19 8
2 0. 3 & [ 8 ( 8 6 H 6 R11
9 2 & ( 8 6 ! & 4. 6 / + . P 8
.% 80 4 G
0. 7 = J
(0. 7)n = d P 8 1-(0. 7)n > 0. 8 K
(0. 7)n > R0. 2 (0. 7)n < 0. 2
(0. 7)1 = 0. 7 , (0. 7)2 = 0. 49 (0. 7)3 = 0. 743 , (0. 7)4 = 0. 2401 (0. 7)5 = 0. 16 8 07
∴ n = 5 P 8
19 9
:
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