Introductions To Statistical Thought
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' # % % % / ' % ! . ! 3 & % % # . " 5 ) % # & # / 5 % % ! / $ & / % 5
/ $ & / " # % & / & / # / $ %%& " " ( # % / & / # / $ & " ' ( # / & ( # & & & / % % / & % % & / " / "! ! ( # & & & / / % / & & ") 3 % /P%& % ) `(θ) % y = 40 ! & / ' 5 / . . % ' / + % / . . ' / # * '3 / / / * *+ # $1 " % / %& % % & .% " / ' (λ, θ ) % *+ # ! 3 / %& % & % # # / # / ! ! . # / % & 3 # # # % # % # (n, .1) !3 / + # 3' ) - / / 5 - / / / %% λ n = 1, 4, 16 3! " - / / / % λ n = 60 33 " - / / /% . . 35 /% ) " & / / % & " " / % / & / n = 0, 1, 60 ) " ' /% % (100, .5) / & 5 " /% (100, .5)% / / , (50, 5) / & 5 " ! 1 + # % / &% ## t 5" / 53 " 3 , & # %# " ) # # & / & % 5) " $ & % + # % % " 53 / & # / "
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+
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( // & & / / & / # / &% / / + & & % & & / / / % & / # & / / % / / # % # & / %& / / / + 1 & & & & & & & % # # % / / ( - // % & & & / & # & / &/ / / # &/ % + &/ # %
& % & % # 4 4 & % / / % # & # / / ; :; 9 ; % 2 ## / / + # / / 4 ( & & & % ## + - / + & % 1 / / & +# & # + # & /# & / & &
+
+
% % /F !"#! - (X , F ) %& & µ : FX→ 1[0,1] / 2 - / µ % & "! $ #%& % F µ / # & & %& / ( # % µ & µ(∅) = 0 ∅ # µ(X ) = 1 - / " % A / A / µ(A ∪ A ) = µ(A ) + µ(A ) % /
" / % & *+ 2 ## / + - % & # # / % " & / & & µ & X / ' "&! ! X - µ / &% %##" 2 / -% F & & # ( F& % # (X &% & & & % X . / % ) . % 55 ) / 2 + / # / # & X & F # F &X
1
2
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1
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2 + # X % & # % + # / / # / + / # + # - / µ + / / / ( % & #& {1, 2, $3, 4,% 5, 6} / & X ≡ {1, 2, 3, 4, 5, 6} 2% / &/ µ({1}) = µ({2}) = · · · = µ({6}) = 1/6 ( % # & & &
µ({1, 2}) = 1/3 µ({2, 4, 6}) = 1/2
% # / µ(2)%- µ(5) - . µ(i) = 1/6 & # /# % & / + µ(i) = 1/6 & % µ $ & & /% # P & + # - & / & / / ( •
/
• P(1) • P[
/ ]
( & + + # / ' " # % & • µ({1})
% % #
% # & /% #
/
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!" #
% $ & ' ( ) * + ,, - ' ( . / " 0 ,/ ' ( 1* * ! '/ " 0" + " ,, ,/( - ' 2 3 ! ( 4 ' ! ( 5 ! ! * ! " 4 ! 6. 1* 4 * 6. 7 " . * 6. * 4 7
"
! ! ! * " ) ! P( ) ! P(
2 3 !
) = P(
+ ,,
) = P(7) + P(11).
! * ," ! d1 - X ! d 2 * , 6 *
d1
X
d2
(6, 6) (5, 6) (4, 6) (3, 6) (2, 6) (1, 6)
(6, 5) (5, 5) (4, 5) (3, 5) (2, 5) (1, 5)
(6, 4) (5, 4) (4, 4) (3, 4) (2, 4) (1, 4)
(d1 , d2 )
(6, 3) (5, 3) (4, 3) (3, 3) (2, 3) (1, 3)
(6, 2) (5, 2) (4, 2) (3, 2) (2, 2) (1, 2)
(6, 1) (5, 1) (4, 1) (3, 1) (2, 1) (1, 1)
1* * " ! 6 0 * " * & " P(d1 , d2 ) = 1/36 (d1 , d2 ) ! + ,, P( ) = P(6, 5) + P(5, 6) + P(6, 1) + P(5, 2) + P(4, 3) + P(3, 4) + P(2, 5) + P(1, 6) = 8/36 = 2/9.
5 . ! ! ! ! 0 * ! 5 *3 * " " " ) " * ! (6, 5) (5, 6) (1, 6)
! *
*+ # & ##% & % # # % / / / - / # # / & & % # & & .# # & / # # # # + # %# # & ( & #& & & 4 & &% / *+ # & 4 & # & # & / *+ # 7 ; ;9 5 ! * " . ! 5 * , 5 4 & # " * * * * " #3 4
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+ ,, 1 ! ! 5 ! # , , ! # ! × 1000 222 ! ! ! ! (2/9) # ≈ * ! 5 ! * * * ! ! / , ! ! ! ! /// ! ! ! * ! 5 ! . * ! ! ! ! ! 1* ! " ! ! 5 . 3 5 ! * !
4
. % / # & % & & X
4
% # % # $ / - & % & . / X .# + # ' "&! !
4
89 # & %
% # %
8; ; / # ; #& % # & & ;
# &/ / / # % & & Y% #/ / /
& # / / / &/ / &% %##" $ + # / ( / # Y $ &
# &# # / & % # / #
p(y)
%
+
3
y
$ & /% % # / '!%" $ #%& % '$ ( ( & & /% /# / - + / y = 0 % # & y # & 6
)
# / /% & % & + - & / # p - π % f /% $& / /%-p(50) π(50) f (50) # & / & & $ +y =# 50 ! # &
P[Y < 60] =
Z
60
p(t) dt.
* /% # & % p(y) ≥ 0 % y R p(y) dy = 1 / & - % p(y) < 0 ( R / p(y) dy < 0 P[Y ∈ (a, b)] = ( / % / & P[Y ∈ (−∞, ∞)] = R & & & & / # % ( & a] = 0 a∈R 0
∞ −∞
b a
∞ −∞
p(y) dy = 1 P[Y = Y
(a, b)
P[Y = a] = lim P[Y ∈ [a, a + ]] = lim
Z
a+
pY (y) dy = 0.
& -% &# a < bP[Y ∈ (a, b)] = P[Y ∈ [a, b)] = P[Y ∈ (a, b]] = P[Y ∈ [a, b]]. % / & ( & & 2 / / = # # & '!%" - # 2 / / & # !% $ % ! 2 / - % / % ""# 2 &/ / % / & & # * & % & # → 0 % / & / $ + / &# a Z d d P[X ∈ (a, b]] = f (x) dx = f (b). →0
a
→0
b
db
db
X
a
X
5
. # - d/da P[X ∈ (a, b]] = −f (a) . # %# % /% % % / & - &$ + # & -/ # / %% # ( Y / & $ &
Y
/ & %& / / # / % # Y # & & . /% p # &
X
p(y)
0.0 0.4 0.8
0.0
0.2
0.4
0.6
0.8
1.0
y
$ & p %
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%
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# / / / # / 1 & & % / - / # / // & / % # / % & # $% & #" # % (# & % # / & & , 1 ' , & / . / #/ / & /# *+ # #% / /% f f & /& / & ( / # % & # /# / & % ##f ( # & & # - % / & &# #& / / / # f & # / # % f 2% & & # // # # / # // / & & % # f & # %f / # / % & / # % $ & " & & # % % / ( # 4 / - 4 + -% ( & # / / # )! 5 5 2 / / , # / / & & (3.1) / & / . " 2 # # % / & % %& & (3.1) / & - & / + +%&#& 2% % # 1 - % / & & %/ / ( &# % / / # & / / & ( & # & & / / &
◦
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5
6
7
8
9
10
11
0.00
0.10
0.20
temperature
0
2
4
6
8
10
12
discoveries
$ & " / , # # % #& / # / & &
◦
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◦
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%
$
$ $ % $ $
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$
,
% 4 & ! !" 4 # & & &
%, 1 / • % # / # & / & / / & / % # & # & # & / % & & • $ ' / 5 & • / / &/ '' / '! # # $ & + •
&/ # / % & & # & . ' •
%/ / # /% . / # / / & // + •
•
"
# / 2 % % % # - Z = g(X) # & / /# / & p X = h(Z) ! X ! %'
•
Z
%# / #% - % + / & g # p ( %& g &- / #
X
! '$ ! ! ' ! pX g !% ! % % & % ! ! $ #%& % %' '! %! !% ! '$ Z = g(X) $ "
Z
$
2% g
pZ (b) =
−1 d g (t) pZ (t) = pX (g (t)) dt
%&
−1
d d P[Z ∈ (a, b]] = P[X ∈ (h(a), h(b)]] db db Z d h(b) pX (x) dx = h0 (b)pX (h(b)) = db h(a)
% g / % + /% p (x) = 2x ( & - & X / # % # ( & Z = 1/X p (z) X = 1/Z 2 / dx/dz = −z ( % -
(
(/ #
X
−1 d g (z) 2 1 = − = 2 pZ (z) = pX (g (z)) d z z z2 z3 −1
Z
∞
∞ 1 2 dz = − 2 = 1. z3 z 1
P[X ∈ Ix ] P[Z ∈ Iz ] x = (Iz ) (Ix ) (Iz )
pZ (z) = 2/z 3
& / x - z2 & ( x / I I " (I ) ≈ p (x) |h (z)|
%& # z = g(x)+ 1 / $ & ' ( / g & / z
pZ (z) ≈
Z −2
1
& % Z - % # ∞% . 1 (1, ∞)
1/
z
X
0
x
' + # * & " + %I /I ( / % / & % R 2% g / & g
x
z
Z=g(X)
z
x
$ & '
%
# & &# . - % +% # / - / / & / ( θ # & P( ) = θ & θ % % %2 # # & &# % / / -% / + & θ / / θ- & + # - P( | θ) P( | θ = 1/3) / ( % / . P( | θ = 1/3) / θ & % " / P( | θ) / / ( θ
#
/ 2 / % % ( & % θ / / Θ & ( % µ
& / / . & / / / - / θ! ! ( % θ & θ $ θ % # # & # /& -
θ
( (
P( | θ = 1/3) = 1/3, P( | θ = 1/3) = 2/3, P( | θ = 1/5) = 4/5
{µθ : θ ∈ Θ},
% / / & % & %! & $# & %&
# % # % #& (% # &
/ % % & & % & 1 • / • . / % & & / • & % & ( & # % & & & • .& / % . # + # # # * 89 1 &/ # + / / *
/ / * # ; & / / # & / * .
%
! / & & # + # / # * / &# % & & / / N # & % % / & / & & / / & & " X / X ∼ " p ' " # !' θ( #& ∼ & / " ' " # !' (N, p) " % " ! % &' " # % .# # & # N% p & # + # % N + / / -θ # % - / & # / # #& & θ)+ / # && / / X ∼ (θ) N =/1 / & X % % ## + # / &% # /- X & / & N % % % # ( P[X = k | θ] * & ' /& k N . N ' P[X = k | θ] = θ (1 − θ) k ( # / % &! & !% / / N k % & % k / # = & & % # /% # % / # 2 k = 0 k = N - 0! / / $ & & N # % N ∈ {3, 30, 300} / p ∈ {.1, .5, .9} ; ;9 5 # ! * 6 # , , , /
N k
k
N k N! k!(N −k)!
N −k
* # * ! 2 3 ! 5 # ! ,
2 3 ! 1 !
! "
/ 5 ! * 0
! ! * !
! * ! 5 * ! " X * ! # , , N6 = 4 p = 2/9
X ∼
!
(4, 2/9)
2
0
1
2
0.6 0.0
probability
0.35
3
3
0
1
2
N = 30 p = 0.1
N = 30 p = 0.5
N = 30 p = 0.9
30
0
10
0.00
probability
0.00
probability 10
30
0
10
30 x
N = 300 p = 0.1
N = 300 p = 0.5
N = 300 p = 0.9 probability
0.00
probability
0.06
x
0
$ & #
0.06
x
0.04
x
200
200
0
x
3
0.20
x
0.15
x
0.20
x
0.00 0
N=3 p = 0.9
0.15
probability
0.6
1
0.00
probability
0
probability
N=3 p = 0.5
0.0
probability
N=3 p = 0.1
0
3
0.00
200 x
)
5 * ! 2 3 ! ! 2 3 !
P[
1
] = P[X ≥ 1] 4 4 X X 4 (2/9)i (7/9)4−i ≈ 0.634 = P[X = i] = i i=1 i=1
+ * &
P[X ≥ 1] = 1 − P[X = 0],
/ 4 ( $ ## / # & & & ( # & * & & / # $
/ # % ( / & % X% ( ; & &
& #
% ## & % % ( / # % & / - & & • * # /# /# • ( &/ • .& / ""% % 5 & # . # ( & # % &/ & .# +# & %# %
( / #
; & & # & / ; % & /% ; & & &
1
$ / ## %
5
& # & % & % % / λ &# % ( 2# (λ) & #/ # & / / X X ∼ % & #& + # & & / & / & /# ` / % & / ` ( + #/ /- X & * & ! & ! λ e P[X = k | λ] = . k!
% # # % & / # & / *+ # ' & & % & λ / & 9; ; 5 ! . 6 "
1
2
k −λ
" ! .! ! . " ! ! * 1 ! ! * . ' 3 ! ' *! *
6 * * * " * * * " * ! . . 5 ! ! . ." ! ! * ! ' ! ( ! . . ! ! ! . / / *! ' ! * 6 ! " λ ! " . ! .! * 5 λ .! * P[X = 3 | λ] .! * λ #
" λ
X=3
P[X = 3 | λ]
* .! * & # " λ
13 e−1 3! 23 e−2 P[X = 3 | λ = 2] = 3! 3 −3 3e P[X = 3 | λ = 3] = 3! 43 e−4 P[X = 3 | λ = 4] = 3! P[X = 3 | λ = 1] =
≈ 0.06 ≈ 0.18 ≈ 0.22 ≈ 0.14
1 " .! # * ! 3 .! .! ) ! λ = λ=1 λ=2 λ=4 *! * 5 - ! ! & ! , P[X = 3 | λ] λ . -
# 5 ! = 3 | λ] λ = 3 P[X " ! " . ! , 5 - ! .! * * ! ! # ! λ
0.00 0.10 0.20
P[x=3]
λ=3
0
2
4
6
8
10
lambda
$ & ! P[X = 3 | λ] %&
%λ
$ & ! / & /
%
$ $ $
,
% #/ % & / % # / & & &
$ % / & • & / % # / & / 2 - ## / • + + -& + / + •
/- / / % λ/ + # & 0& / / & / 4
$
2 % # / /# / % # . # + # & & # /
# & + &
# & + & %%% 89; # % /
/ /
$ / % / / & % $ & ! & % & &
X
/
2 %+ # + / # - # / / / . / / 0 / x 1 & %& /% % & & & x=%!% '!%" % x > 0. 1 3 p(x) = e λ ' " # % / ! ! *+ X$ " % 3! %!% λ X ∼ %
+ / / % (λ) & & λ
8 6 0
2
4
p(x)
0.5
1.0
1.5
x
$ & 3 *+
/
lambda = 2 lambda = 1 lambda = 0.2 lambda = 0.1
0.0
10
x −λ
2.0
$ & 3 / & / %
"
%
% $
%
$ $
$
+ / % / % / / & λ ' & %x ,+ ' & % λ • % / # ( / & / p(x) % • λ# + # % ## ++ / 2 / - x ( ' ' !
+ /% ( & # 4 x • & /% # & 4 && % # 4 & / ( & % *+ - 4 *+ ( # + & # + / ( • / # % # % 2 & - & % & # % - & & % / 4 & / / / ( & • x / % & % % / + y & 3 2 & / -' % & & ! !%" % $ & / 2 # / % / % • & & •
'
2 % # / /# / Y # & / / . # + # & &
;
% ;
# & & ; # % 9 ; .1( 2 /# + / / % & $ / % & & # % %# /% . /% & /% & # / // / # 1 & %& /% % & & '!%" 1 ) p(y) = √ e ( ). 2πσ ' " # % " !% %' " %'' '! % / Y µ , , # / % ) $ & σ Y ∼ (µ, σ) /% % / / & (µ, σ)/% 1 & / & & µ% % µ
/- σ / /% & σ # / /% # σ & $ & ) / & / %
$
y−µ 2 σ
− 12
%
mu = −2 ; sigma = 1 mu = 0 ; sigma = 2 mu = 0 ; sigma = 0.5 mu = 2 ; sigma = 0.3 mu = −0.5 ; sigma = 3
0.0
0.2
0.4
0.6
y
0.8
1.0
1.2
−6
−4
−2
0
2
4
x
$ & ) , # /
6
!
# , # /% # % x & & / & # (# & / & # / // # 1 %& & - $ & 5 % # % % // 1
# & 5") 553 & / # - ' / , / /" / # ( /& ( , (5.87, .72) / & # /& , # / / # / % &/ # & #& # / +# *+ # $ & 5 / & /
•
◦
$ $ % $
( # & / & # ( & # % • / / ' & % ( • & , # / % / , # / • 1 & & - & & / & / / 4 %& %
( # / & # % / // ( /# •
$
## / / # # % # , # / % % # #
& & [4, 7.5] &
3
0.4 0.2 0.0
density
0.6
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
temperature
& 5 / # ,$ (5.87, .72)
&
, - # / (
45◦ , 30◦
)
, (5.87, .72) / # 2 & % , # / # / $ &
- / $ & 5 - / # # / $ & # / + / # / # % & / # / % / # , # / 2 & & $ & 5 # & /& /
$ $ $
! ! ! % # % % /, /#/ / -& # " %'' &#% " & / / & && / / / 2 $ & & # ! ! & / /& # # ' & & & %& / /& # ( $// & - # # # % ## $ & / - # & / % 5 / "# & / & & / $ ( & 4/
/ / , # & & &
,
◦
◦
◦
◦
/ 4.5 = 40.1 $ 1 + '" 3 / + % / / / & & / -4 # / &
◦
◦
/ # # - / # & & & 6.5 =/43.7 $ ! / // / 1 #& # + / % ' # / ' ( / ( & + - / %/ # / . / /& % / # & ( / / & & - % / / #
/ / - & , # & #
5
0.4 0.2 0.0
density
0.6
(a)
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
degrees C
0.15 0.00
density
0.30
(b)
39
40
41
42
43
44
45
degrees F
0.4 0.0
density
0.8
(c)
−2
−1
0
1
2
standard units
/ , (5.87, .72) $/ & 11 ## % % % # %, (5.87, .72) # , / (42.566, 1.296) , / 1 # % % # , (0, 1) / 1.296) , (42.566, / (0, 1)
" / % $ &
/& /
$ %
$ $
$ $
$ $
, (µ, σ) / / /# Z = (Y − µ)/σ Z / / & 2 # % / / & Y # µ / & Z ( / & - / % Z- / $ # & p ( #
Y ∼
/ & /
, (0, 1) / 2 - Z ∼ , (0, 1) ( , (0, 1) " %'' / &
6 $ & " % # / / / & & /& # #/ # & /% # & / /
1 2 1 pZ (z) = pY (σz + µ)σ = √ e− 2 z 2π
Z
2 % &/ # / &% & /) /
# % / /% #% 2 / p - & # / / &/ &/% & / # &
◦
"
# / / & 1 / & & / " / / & 2 - / % / # % & . % / & % -% / & / % / . ' # % % # / & / / & / & 4 ! ! ! " " !
◦
◦
! " # " . * ! ! * ! ! * ! ! , + " ! * ! * ! . ! * ! ! " ! " " '6 ! , + . ! ! ! * * * ! ' % $ ( & ! , ,, " ) & # )% " * ! * 3 ! ! , , , ' 5 ! * 3 * - ! ! * ! . ! ! * * 3 & ! , ,/ ! . * , 5 ! & ! , ,/ * & ! , 0 & ! , ,/ * ! 3 5 * * ! * 5 * * ! ◦ ! ! 3 5 * ! ◦ ! 0 ! 3/ ' . " ! 9 *
" . ! * . ( 1 * * !" ! . * & ! " 3 ! ! ! ! * ! ! * . ! ! %! % )%& ! * 6 5 ! ! ' , ◦ ! " * "
35
4 4
"
! !
20
30
40
latitude
50
60
'
−40
−30
−20
−10
0
longitude
$ & /
% *&
/ 1%
%/ % / # & # ( % # ## # ( !% % # - % % & # - & !% % # 8; ( % &; ; # - x , . . . , x # % x ,...,x = 1 Xx . 5
1
n
1
n
n
i
latitude = 45 longitude = −30 20
latitude = 45 longitude = −20
10 5
10
0
0
5
20 0
""
40
60
latitude = 45 longitude = −40
20
4 6 8
4 6 8
temperature n = 213
temperature n = 105
temperature n = 112
latitude = 35 longitude = −40
latitude = 35 longitude = −30
latitude = 35 longitude = −20 15 10 5 0
0
5
10
0 2 4 6 8
15
20
12
4 6 8
4 6 8
temperature n = 37
temperature n = 24
temperature n = 44
latitude = 25 longitude = −40
latitude = 25 longitude = −30
latitude = 25 longitude = −20
10 5 0
0
0
5
5 10
10
20
15
4 6 8
15
4 6 8
4 6 8
4 6 8
4 6 8
temperature n = 47
temperature n = 35
temperature n = 27
$/ & , /# - "& - ' / # / - & / - " - ' &
◦
"'
# # % % ( / / x¯ $ & % / ( / x ,/. . . , x & # / / & ## . # % -# % & %& / % / & $ + # # % /% & " & ) - # $ & # / # . # # # % (3.1) / & " - # # % / %% & / & # # / # / + # % - / (n, p) / & $ # / (30, .5) / & $ & ( % & / / - / ( # % / (300,&.9) - % & / # - & 3 / ! !& !' #! ! !& % / E(X)
X
n
1
; ; 7 8; ;9 % X /% F / /% p ( !% & X %F
-
/#
X
!%
X
%X / % X & &
(P i P[X = i] E(X) = R i x pX (x) dx
% / / ( E(X) % X * & / %# 2 // & E(X) ## # % X µ # % % # % / & / &/ / / & +
%
/ #
X
; 2% X ∼
"
(n, p)
n X
E(X) =
i P[x = i]
i=0 n X
n i = i p (1 − p)n−i i i=0 n X n i pi (1 − p)n−i = i i=1 = np = np
n X
i=1 n−1 X j=0
(n − 1)! pi−1 (1 − p)n−i (i − 1)!(n − i)! (n − 1)! pj (1 − p)n−1−j j!(n − 1 − j)!
( P& & % ( + % # # (n − (/ & # · · · % & # # # (
1, p) & & & & # E(X) $ & ; 2% X ∼ (λ) % *+ E(X) ) = λ. ( / ;; 2% *+ (λ) X∼ = np
n−1
j=0
E(X) =
Z
∞
x p(x) dx = λ
−1
Z
∞
xe−x/λ dx = λ.
2% X ∼ , (µ, σ) / % *+ ( E(X) = µ ) . / # & / / / % / & /# / # 2 $ & - / & / # & #& # &
0
0
"! # % # & & / . & / # / / & & &# % / -& % % # # # / & # % / 6 # $ & / % # % / # % # ## # % / %&! / & - " %'' '! % & ; ; ; ( %&! % # y , . . . , y
%&!
1
(
%
/#
X
Var(y1 , . . . , yn ) = n−1
Y
n
(yi − y¯)2 .
; ; ;99 9 ; ( " %'' '! % % # y ,...,y q X SD(y , . . . , y ) = n (y − y¯)) . ( " %'' '! % % / # Y
Var(Y ) = E((Y − µY )2 )
1
n
1
−1
n
2
i
p E((Y − µY )2 ). Y
% % / / σ σ ( / / / % % ## / & / / - #P / & % # / $ (y − y¯) Var(y , . . . , y ) = (n − 1)
/ / / 1 n % % /# / # & / % % / / # / % # # - & % #& # # #& 7 $ Y " %' ! !% Var(Y ) = E(Y ) − (EY ) (
SD(Y ) =
2 Y
Y
1
−1
n
i
2
$
Var(Y ) = E((Y − EY )2 ) = E(Y 2 − 2Y EY + (EY )2 ) = E(Y 2 ) − 2(EY )2 + (EY )2 = E(Y 2 ) − (EY )2
2
2
% % (/ % # $ #& . // /// / /# / - SD(Y ) / SD $ & ' # & / % / # & ##
% &# # % # # & / . / &5 & %/& # -
2
SD
& / //
& / //
%
% / 2% x , . . . , x & / / & /% / & &% / x¯# - - & . " # & . % % # & / / & . /%
& #/ /
1
%
- % - # & &#
&# 5 % # %
( #& %& & & ( /#
•
# % " % # &
•
"3
/ // # & - $ & ' ± / // # &% / * - # / && Var(Y & ) / Y & & % - / (
n
") ;
Y ∼
(30, .5)
Var(Y ) = E(Y 2 ) − (EY )2 30 X 2 30 .530 − 152 = y y y=0 =
30 X y=1
= 15
y
30! .530 − 152 (y − 1)!(30 − y)!
29 X
(v + 1)
v=0 29 X
/
%
29
X 29! 29! .529 + .529 = 15 v v!(29 − v)! v!(29 − v)! v=0 v=0 29 + 1 − 15 = 15 2 15 = 2
29! .529 − 152 v!(29 − v)!
Y ∼
, (0, 1)
SD(Y ) =
p
15/2 ≈ 2.7
!
− 152
. *+ 5 /
"
Var(Y ) = E(Y 2 ) Z ∞ 2 y2 y √ e− 2 = 2π −∞ Z ∞ 2 y2 y √ e− 2 =2 Z0 ∞ 2π y2 1 √ e− 2 =2 2π 0 =1 SD(Y ) = 1
/ % . *+ 5 / $ & "/ # # ( , / /% % (30, .5) & (0, 1) &
"5
0.10 0.05
+/− 1 SD
0.00
p(y)
0.15
+/− 2 SD’s 0
5
10
15
20
25
30
1
2
3
0.0 0.1 0.2 0.3 0.4
p(y)
y
+/− 1 SD +/− 2 SD’s −3
−2
−1
0 y
$ & , " ( /% (0, 1)
±1
/ ±2 .
(30, .5)
#
' $ & " / & /
4 /
%
$ $
$
.
; ; 8 ; ( /# / / Y
E((Y − µY )r )
/ &#
%
!%
% #
#
y1 , . . . , yn
%
% # % /# /
(yi − y¯)r
// % # %
/ # # #
r
X
n−1
& # // # + &
•
. / &#
•
' / #& 4 & %& # & # / % # # , 4 & / n − 1 / 4 % $ %
$
% %
% %
% % % % %
% %
$
/ #/ % # # / & # / / ( % # (100, .5) / & / %& ( E[h(Y )] = R h(y)p(y) dy P h(y)p(y) h / + / & % h(Y ) / / !%! !' # # & E[h(Y )] & !% ( X = h(Y ) - / p - / % ( R + # - Y /% f (y) = 1 y ∈ (0,& 1) $ E[X] = xp (x) dx / 8 9X = h(Y ) = exp(Y Z) •
(
X
X
Y
1
8 9 7
E[h(Y )] =
0
exp(y) dy = exp(y)|10 = e − 1.
pX (x) = pY (log(x)) dy/dx = 1/x Z e Z e E[X] = xpx (x) dx = 1 dx = e − 1
%& 2% h E[h(Y )] & $ X = a + bY !% E[X] = a + bE[Y ] 1
1
% #
'
$
EX =
& & /
% +
Z
(a + by)fY (y) dy Z Z = a fY (y) dy + b yfy (y) dy = a + bEY
#% ( / $ X = a + bY !% Var(X) = b Var(Y ) $ / % + & & µ = E[Y ]
2
Var(X) = E[(a + bY − (a + bµ))2 ] = E[b2 (Y − µ)2 ] = b2 Var(Y )
. % / #& & % % - / # $ + # - #% % & / / & / / & # A / S / & - ( ( % # % % D Y / A= S= %6 & . % / R N % # /" % .& % / ) 6 & / % / & # ( ) / " / &% ' % & / (
latitude = 45 longitude = −30 20
latitude = 45 longitude = −20
0 4 6 8
4 6 8
temperature n = 213
temperature n = 105
temperature n = 112
latitude = 35 longitude = −40
latitude = 35 longitude = −30
latitude = 35 longitude = −20 15 10 5 0
5
10
0 2 4 6 8
15
20
12
4 6 8
0
'"
5
10 0
5
20 0
10
40
60
latitude = 45 longitude = −40
20
4 6 8
temperature n = 37
temperature n = 24
temperature n = 44
latitude = 25 longitude = −40
latitude = 25 longitude = −30
latitude = 25 longitude = −20
10 5
4 6 8
4 6 8
4 6 8
temperature n = 47
temperature n = 35
temperature n = 27
$/ & ' ,
0
0
0
5
5 10
10
20
15
4 6 8
15
4 6 8
# & # / - &/ - " - ' &/ - " - ' / - / / /
◦
''
$ 1 # ') 6 & ' ! ! ") /6% ( 1
%
/ /#
1 - & p & &. &
! ' / & # .&
S |A
pS | A (Y | D) = 0.80; pS | A (Y | R) = 0.35;
/
pS | A (N | D) = 0.20; pS | A (N | R) = 0.65.
- . % / S=N A=D % # ! & & %( # % )/ # % ! ') % / & %& (A =( D,')S = Y ) % p (D, ( Y ) = .48 / ( & # p (D, N) = .12 p (R, Y ) = .14 p (R, N) = 0.26
#
& ( ( #& / %! -# ' - ! - / " ) - / % & % # % - / & / # (A = ( A=D / . D, S = Y ) (A = D, S = N) A,S
A,S
A,S
A,S
(
pA (D) = .60 = .48 + .12 = pA,S (D, Y ) + pA,S (D, N).
(
/
A=R
(A = D, S = Y )
/ # ( / (A = R, S = Y ) .
S=Y
pS (Y ) = .62 = .48 + .14 = pA,S (D, Y ) + pA,S (R, Y ).
! % & & $ %! ! '' ! % !" $ #!" $ ! ! % #& % #& & / ! (
/#
X /Y
'
fX,Y (x, y) = fX (x) · fY | X (y | x) = fY (y) · fX | Y (x | y) X X fX (x) = fX,Y (x, y) fY (y) = fX,Y (x, y)
'
.# # / / /# / / # # / 1 / # # f /% f / / / f f (& % +# # /& -/ 2 ## - /# / # / # / ( % / & ( 5 & % % & / - 5 % % & & % & -" % / 1 & / # & . &/ & && / & ( # % %/# - U - % / & % / T - % & % U =1 & / U = 0 T = 1 %& T = 0 % / f (1 | 1) f f & & f & f / f y
U,T
(
&
X
U,T
T |U
fU,T (0, 1) = (.7)(.1) = .07 fU,T (1, 1) = (.3)(.9) = .27
fT (1) = .07 + .27 = .34
fU | T (1 | 1) = fU,T (1, 1)/fT (1) = .27/.34 ≈ .80.
fT (0) = .63 + .03 = .66
/
fU , fT | U −→ fU,T −→ fU | T
fU,T (0, 0) = (.7)(.9) = .63 fU,T (1, 0) = (.3)(.1) = .03
U |T
U
U |T
X |Y
Y
Y |X
x
'!
! " " ! ! ( . /
3 3 3 " " ' / ( 6 &
T =0 T =1
U =0 U =1
2
/ - % & & % / 2% /5 & # & %) % # - 1 " / # & 3 & 1 & 3 & & % #& & % 3 . "' & # & 3- ) & % # # 2 % # ( % # # / # # / / / & - / & # # / # & &*+ # ! % % - / - / # & & / & 9; # , ! . # ! *
! ! . . 2 5 2 & ! ! ! " ! * 6! 4 N N ∼ (λ) * . ! . . λ>0 # * ! . . ! * ! . . 6θ! ! . . * X
! . . * 5 (N, θ) 5X∼ & ! , , .! * X) * & ! , , (N, " fN,X !!" ! . ! & # "
fN,X (3, 2) N =3 X=2 5 # - ! ! ) 2 " * # " . 5 . * ! N = 3 " . " "
(N = 3, X = 0) (N = 3, X = 1) (N = 3, X = 2)
'3
...
...
0
2
X
4
6
0
1
2
3
4
5
6
7
N
$ /& # # & % N / X & &
(N = 3, X = 3)
&#
%
/
6
fN (3) = fN,X (3, 0) + fN,X (3, 1) + fN,X (3, 2) + fN,X (3, 3)
5 4 * ! −λ 3 N f λ /6 N (3) = P[N = 3] = e −λ 3 . * ! 5 * e λ /6
5 .
X
3 (1 − θ)3 0
3 θ(1 − θ)2 1
3 2 θ (1 − θ) 2
3 3 θ 3
5 −λ 3 7! e λ /6 5 . #)% )% " / N = 3 5 f (2 | 3) X|N 5 )
P[X = 2 | N = 3]
e−λ λ3 e−λ λ3 (1 − θ)3 fN,X (3, 1) = 3θ(1 − θ)2 6 6 e−λ λ3 2 e−λ λ3 3 fN,X (3, 2) = 3θ (1 − θ) fN,X (3, 3) = θ 6 6
fN,X (3, 0) =
')
1 " e−λ λn n x fN,X (n, x) = fN (n)fX | N (x | n) = θ (1 − θ)n−x n! x
" * * ! * ! ! fX ! & ! ! "
& ! , , 3 x f X (x) 5 " ! ! X = x fX (x) ≡ P[X = x] '
X
fX (x) =
n ∞ X
∞ X e−λ λn n x fN,X (n, x) = θ (1 − θ)n−x n! x n=x
e−λ(1−θ) (λ(1 − θ))n−x e−λθ (λθ)x (n − x)! x! n=x
=
∞ e−λθ (λθ)x X e−λ(1−θ) (λ(1 − θ))z = x! z! z=0
e−λθ (λθ)x x! * P =
5 ! ! ! * · · · = 1 z * 4 ! 5 - ! −θ)) * 4 (λ(1
! ∗ 6 4 ∗ ∗ (λ ) λ = λθ X ∼ (λ 1 . ! ! ! 5) z = n−x
$
& & / / #
pX | Y (x | y)
/ / /& * & ' p (x, y) (x, y) = p (x)p (y | x) = p (y)p (x | y)
- /
X,Y
pX,Y
X
pX (x) =
Z
∞
Y |X
Y
pX,Y (x, y) dy
pY (y) =
Z
X|Y ∞
pX,Y (x, y) dx
# % / % %/ # 2 / % (X = / x, Y = y) & $ X = x & Y = y -& ( % X = x & p (x) × p (y | x) (
−∞
X
Y |X
−∞
'5
& # / / Y = y & & / / - % & & & X =x / # & & & & %& & $ Y = y & X = x & ( # # % /# % / 0& R 2%R A - P[(X, Y ) ∈ A] = p(x, y) dx dy (x, y) / / A ... & % / /# / / % / #0& / % / p% & / p # / . / & #& / B⊂R R /- P[X ∈ B] = p (x) dx
& /
Y
X
B
Z Z
P[X ∈ B] = P[(X, Y ) ∈ B × R] =
A
X
A
pX,Y (x, y) dy
dx
# p (x) = R p (x, y) dy 1 + # # % #& 1 / & # / / # & / % X ( #& & % # & / Y # % & / W = Y − X . & / p (x, y) = e 0<x
X
B
R X,Y
R
X,Y
"
p(x) =
Z
p(x, y) dy =
Z
∞
−y
e
x
" ! % '!%" $ p(y) =
Z
Y
p(x, y) dx =
" ! &%' % '!%" $ p(x | y) =
∞ −y
dy = −e
= e−x
x
X
Z
y
e−y dx = ye−y 0
!%
Y
p(x, y) = y −1 p(y)
−y
'
" ! &%' % '!%" $
" ! % '!%" $
X
p(x, y) = ex−y p(x)
p(y | x) =
!%
Y
W
Z ∞ Z x+w d d P[W ≤ w] = p(w) = e−y dydx dw dw 0 x Z ∞ x+w d = −e−y dx dw 0 x Z ∞ d e−x (1 − e−w ) dx = e−w = dw 0
$ & !# &% # * & ( $ & / /# # # % %& 2& & -% +% # -% % & & (X = x - Y # # x% ∞ - # % # #2 - % & & / % # & / y X ∈/ (0, y) % % / & . % Y ' % / % % X &X % # Y . / % & / % / /(0, y) / % Y / % & X 2 - - Y > X% / /X / / + . . / " " Y & % / % % % # & / / # / & / & & ( - # # /% / /# X & / & +
(#
/%
E(X) =
&
Z
xp(x) dx.
& # % X / # X / + (X, Y )
E(X) =
Z
xp(x, y) dxdy.
(b)
0.4
p(x)
2 0
0.0
1
y
3
0.8
4
(a)
1
2
3
4
0
2
4
6
x
x
(c)
(d)
8
10
0.0 4
6
8
10
0.0
1.0
2.0 x
(e)
(f)
y−x=w y−x=0
0
0.0
2
0.4
0.8
x=1 x=2 x=3
3.0
6
y
4
2
y
0
p(y|x)
y=1 y=2 y=3
1.0
p(x|y)
0.2 0.0
p(y)
2.0
0
0
2
4
6
8
10
0
1
2
y
3
4
x
# $/ & !% # %/ R % (X, Y) 9 / %/ % X Y % % / / & X Y Y Y X % % & X W ≤w
2
(%
% #& - % & - & -
&
p(x)
/
X E(X)
/
! 3 (x | y) %(& % y - / g(y) (# +# // # $ * & & # !x- p R g(y)p(y)dy - / # . # - | Y ) %& & % Y E(g(y)) - h(Y ) ( # // # * & 3 RE(X h(y)p(y)dy / # & E(h(y)) - 9; ; ;9 # , , ! " ! * * Z
pX (x) = pX | Y (x | y) pY (y) dy = E pX | Y (x | y) Z Z E(X) = xpX | Y (x | y) pY (y) dxdy = E (E(X | Y )) .
X |Y
(N, X) ! . . . # , ! ! *
! 4
N∼
(λ)
! , ,+
E(X)
X |N ∼
(N, θ).
!
E(X) = E (E(X | N)) = E(Nθ) = θE(N) = θλ.
; ;9
# , ," , / , 0 ! * # , ! *
X
! ! P[
' ! !
*
*
( 0 X= 1
-
] = pX (1) = E(X).
pX (1) = E(X)
(
Y
! * 2 3 !
"
! , ,+ E(X) = E (E(X | Y )) = E(X | Y = 2) P[Y = 2] + E(X | Y = 3) P[Y = 3] + E(X | Y = 4) P[Y = 4] + E(X | Y = 5) P[Y = 5] + E(X | Y = 6) P[Y = 6] + E(X | Y = 7) P[Y = 7] + E(X | Y = 8) P[Y = 8] + E(X | Y = 9) P[Y = 9] + E(X | Y = 10) P[Y = 10] + E(X | Y = 11) P[Y = 11] + E(X | Y = 12) P[Y = 12] 2 3 1 +0× + E(X | Y = 4) = 0× 36 36 36 4 5 + E(X | Y = 5) + E(X | Y = 6) 36 36 5 4 6 + E(X | Y = 8) + E(X | Y = 9) +1× 36 36 36 2 1 3 +0× . + E(X | Y = 10) + 1 × 36 36 36
6 - * 5 ! E(X | Y = y) y = 4, 5, 6, 8, 9, 10 * ! = E(X | Y = 5) # * w z " * ! % ' * (" ") ! z = 5 ' * (" )"" & % ' * ( 5 * z=7
z
w = 1 × 4/36 + 0 × 6/36 + w × 26/36 (10/36)w = 4/36 w = 4/10.
'
* ! * - E(X) =
3 3 4 4 5 5 6 5 5 4 4 3 3 2 + + + + + + + ≈ .493. 9 36 10 36 11 36 36 11 36 10 36 9 36 36
2 . * !
2 % & %& / # & / % /# X / Y ( 4 / /
/ + # 2
'
%
(
/ 3 % # / / / % / #& . / & %4 / 3 / & / %
$ /& % & $&
/ / % / %
2 /
% - # + % % / &( " % # # % # + ( (i, j) % / # + + $ / + # - / & j i & + % # ## # & % & %&! / &! % ; ; ( & %&! % X / Y •
# ( & -
Cov(X, Y ) ≡ E((X − µX )(Y − µY ))
2 # / & 4- # &+
3.0
4.0
0.5
1.5
2.5
6.5
7.5
2.0
2.0 2.5 3.0 3.5 4.0
4.5
5.5
Sepal.Length
5
6
7
Sepal.Width
0.5 1.0 1.5 2.0 2.5
1
2
3
4
Petal.Length
Petal.Width
4.5
$ & 3
6.0
7.5
1
3
5
7
/ / % /
%
!
/ ( #/ # & # # # & & &
# #
/ / # (
* # / #& / 2 % - & & % % # ! X %' Y ! %' !" !% Cov(aX + b, cY +
d) = ac Cov(X, Y ) $
Cov(aX + b, cY + d) = E ((aX + b − (aµX + b))(cY + d − (cµY + d))) = E(ac(X − µX )(Y − µY )) = ac Cov(X, Y )
# % Cov(X, Y ) / % / / X % # & / 1 # Y & / & & # ## & & # & ; ; ( &! % /Y X (
# &
Cor(X, Y ) ≡
(
Cov(X, Y ) SD(X) SD(Y )
3 - /
# & # / / -& / / & / ( # ! # & # & ! X %' Y ! %' !" !% Cor(aX + b, cY + d) = Cor(X, Y ) $ . *+ ' / # & % % # & &/ & / / ( - # & # $ & )/ &/ / ( /& % & # & + & & / & / , & % % & # / & ( %( && ## /% / / & % % / % / 5 !# / / / # / ( & & % # % 1 % &# / # / $% - # / &/ & & # / # ( // & % / & - / & # & /# & # # /# - & ; ; ( / # / / %'! !% X Y '!% % -% % 2% / / / p(X | Y ) = p(X) / '! !%'!% & Y X Y 2% X / Y / / % & p(Y | X) = p(Y ) % /# / ( / 2% / / / X Y & 2% / / /Y /% / X ⊥ Y X 6⊥ Y X & 1 - X ⊥Y ( Cov(X, Y ) = Cor(X, Y ) = 0
)
cor = −0.9
cor = 0.3
cor = 0.96
cor = −0.5
cor = 0.5
cor = 0.67
cor = −0.2
cor = 0.8
cor = −0.37
cor = 0
cor = 0.95
cor = 0.62
$ & )
5 %
/ ( & & & / / % / / # + / & & & & X / A = 1 % X ∈ {1, 2, 3} / A = 0 % X ∈ & # % / / %' & / & & {4, 5, 6} A % % / ( & A=1 (X). %' & $ #%& % 1 (X) / 1 & B = 1 (X) - C = 1 (X) - D = X / E = 1 (X) A / B / / P[A] = .5 1 (X) & & P[A | B] = 0 D / E / / & P[D] =% P[D/ | E] /= .5 & / P[E]/= P[E/ | D]- = 2/3 & & & & A B '5 X / Y $ # + %# # - / / % & W = Y − X % & / p(x | y) = y & p(x | y) ( / / / . # - p(y | x) / / x # & y/ X 6⊥ Y # X & & & Y / / / 1 & X W
p(x, y) = p(x)p(y)
{1,2,3}
{1,2,3}
−1
{1,3,5}
{4,5,6}
{1,2,3,4}
{2,4,6}
{1,2,3}
p(w | x) =
d d P[W ≤ w | X = x] = P[Y ≤ w + x | X = x] = e−w dw dw
/ / / x ( % X ⊥ W # # &/ & X & & W < 9; ; ;9 # , " , " , + ! * ! 6!
. ! * ! * ! . . " " 1* . k N1 N k . " ! 4 1 " Ni ∼ λ * . 1* " * (λ)" ! * λ * λ ! ! !
" ! " ) " " ! Ni
λ
p(n1 , . . . , nk | λ) =
k Y i=1
p(ni | λ) =
k Y e−λ λni i=1
ni !
P
e−kλ λ ni = Q ni !
!
! *
! . 1 ! λ . *" * " * λ * ! * ! * ! ! 1 * " " λ " * ! * ! .! * N1 , . . . , Nm " Nm+1 , . . . , Nk λ1 * " Ni
Z
p(n1 , . . . , nk ) =
p(n1 , . . . , nk | λ)p(λ) dλ =
Z
P
e−kλ λ ni Q p(λ) dλ ni !
6 " ! Ni 1 " ! . ! " ! Ni . λ
/ % # / & &
#
- *+ # - + # % # & % # & / - #& & & ( + - / / + # / / & +
% % % # #/ #/ & # & && &/ /- & # / % # # # / # & $ - *+ # # 3
& 1 & & & / / #& *+ # *+ / & # / # & / & # + / & # # #% & µ ≡ E(g(Y )) % ( % # & &% & y # # ( & Y n & &
µ ˆ g = n−1
X
g(y (i) )
(j)
g
. #& / # + / # Y % # %& # # Y #& / j
# %µ 2 %
g
. /% # µ
X
,&# &
g(y (i) ) = µg .
#& # /# & # & / % + . & % ( & /P[Y ∈ S] ## / S X ≡ 1 (Y ) P[Y ∈ S] = E(X) & # # % &% % S = n X x . & *+ # & # % ; ;9 # , ! * * 2 4 ! !
/
lim n−1
%
!
n→∞
g
S
−1
(i)
% % % %
%
% %
!
%
4 # % % 2% # ( •
•
#
(# + # / % # . '5 *+ & & ) & % # %- & %& # 1 & % # % µ 2% # &% / / µ ˆ # # Var(g(Y )) Var(ˆµ ) = n Var(g(Y ))P # /pPn ((g(y) − µˆ ) ) / SD(g(Y # / )) / . n ((g(y) − µ ˆ ) ) SD n & #% ( # - % / n - % +# - / & .# % - & #& # & - &# + % # % & / & & SD ; ;9
−1/2
g
2
2
g
−1
g
g
−1/2
! ! # , , 5 ! * " ! * ! X ! * ! ' # , * ! θ
g −1
θ ≈ .49
" !
!"
( θ
, θ) (θ)(1 − θ) p SD(X/ ) = (θ)(1 − θ)/ " 2 5 * " θˆ = X/ ∼ (θ, (θ(1 − θ)/ ) ) ! ! = 50 " / " , " 5 ! , ! = 50 " , = 200 " , = 1000 5 ! # & ! , , 1 & ! , , ) ! " ! # . 5 # * 50 * , ) ! ! ' =
X∼ Var(X) =
(
1/2
ˆ
! #(" * ˆθ * ! ' ! * #(" θ * ˆ * ! 0 ' # * ( 1 " θ, ˆ * θ = 200 ! ! * ! " , ˆ * ! ! * ! 5 * * !θ * =* 1000 .5
* ! *
* * ! " ˆ SD(θ) SD * * ! / 6 * & ! , , * *! * #
4 /%
#&
/ $ & 5
!'
0.3
0.5
0.7
50
200
1000
n.sim
$ & 5 # &
% θˆ % - -
$
$
% # # + # ( + % /# / # % & / & # # + •
# •
1 $-
•
*+ /
/ / 2 /& # // ( & % / ( / #/ % / / + / / /
! % & % / 1 # " !" ( % + #/ % / . *+ # " & % + / %& +
. #& # # & %& / / + & # # + # / + # - / % - & & / $ - " % & & / # # 1 % % / / ( & # % / % &// 3 ( // & - / * % & θ = % / / & θ = % & / & θ= 1 # ##/ / / # θ / / / / & ( & 6 / # # & / * # % / &/ % θˆ = 6 / # # & / * # % / & θˆ = % / / / & θˆ =
I
G
I
θˆ2 = .3θˆG + .7θˆI
" 6 / # # "
1
G
θˆG = θˆI = θˆ3 = .3θˆG + .7θˆI
/ 3 / / * # % / & % / / / &
!!
/&
+ & & / - #& # # & -2 # & # # & & & % θ - θ / θ & /& / #& & # 4 / % #&
$
% $ %
% $
$
$
I
% $ %
G
%
%
!3
%
$
$ %
% %& # + 2 % / -
& %/ #% # + ( - & # &# % / ( + - / & $ & & / & ( + + # #& + # & 7 5 ! * # " 2 "
•
. ! 2 * ! ! *
5 " ! & ! , / , 5 . * # 5 * 2 3 2 ! ! * , , + 5
! ! 5 *
! * * *! 2 ! 2 * !" 2 .! ! ! ! 2 2 ! 1 2 ! ! * 2 # 5 ! ! . 2 " 3 ! # ! 1 !"2 * !
4
!)
0.35 0.50 0.65
1
$ & # & /&
2
% θˆ & /
3
/& %
! # * ! * 2 2 ! . * ! ! ! ! ! . " " . ! " # . 3 3 " * " ' " !! " 2 ! 2 " ! 2 " 6 2 # 2 ! 2 ! !" 3 * ! # ! . ! 5 " , * . 6 * ' ( ! ! & 2 " !! % # $ !% . . ' 2 $ ' #
& 2 '
. !
!5
(b)
0 −2 −4
320
co2
340
resids
2
360
4
(a)
1980
1960
1980
Time
Time
(c)
(d)
320 2 4 6 8
$ &
340
co2
−3 −2 −1
0
cycle
1
2
360
3
1960
12
Index
1960
%
1980 Time
2
&
3
- ! 3 5 2 /3 * * . ! 0 " / & * 3 ! * 2 / . " . 2 / ! ! & 2 !
- ! 3 ' & 2 3 ' ! * ! *!3*! " 3 & 2 & ' * 2 / #
33- ! 6 * ! * 1 & 2 # " 2 ' 2 5 . * 2 ! 2 2 ! * 2 2 - ." . # 2 3/, ! . 4 ! & 2 " 2 ! * ! * ' ! 2 2 .
" * " ) # ! & 2 2# 5 - . & 2 ' ' ! " " (" 6 ' "' . ( 2 * ' ' . ( * ' 3 , " 3 ! & 2 # * " ! # '
. ! . " ! & ! * # ! *! . . * ' ( ! 1 ! " 3 # ! . * ! . . ! # 5 3 ! ! ! 5 * " * " . * # # , 5 # ! * " . 0 ! . 2 ! 5 # 2 ! * , 1. 3 - , ! & 5 . ! * " ! * # " . $
3
/ 5 ! , * # 5 *
0 5 . ! * % % 5 ! * * . * ' " " ( 5
! # " " - 5 .
! # ! ! 5 ! ! $ $ '
* , . 5 . 5 - .! $ . '
1. # ! . *
/ . * 5 .! $ $ 5 ! ! $ , " , " , ' ( , , ! . ! * $ $ & " * $
( , 0.1 ∗ ($ − 1)) ! $ " *
+
!
* # . 6 ! 5 ! ! 3 ! 5 ! * ! (1,
)
# ! ! * & - . !
% %
3
& ! ! !3 * $ $
!
!
! ! ! 1 !
$ $
, 5 ! * !
)
$ "!
- * * 6 6 # * ! 3 & ! ! " . 3 " & ! , // " * ! 5 *! * & 2 # ! & " ! * '! # ! . * !
% $ $
3"
1.0
1.5
growth rate
2.0
2
4
6
8
10
year
$( & + . %/y / /
# & % $1 * # / / / /
+ # ( x + # # # i & /
3'
$ % $ % $ $ $$
$
$ $ $
$
$
$ $
$
% $
$ % % $ $ % $ % % $ % $ % % % $ % % $
% % $
% $
3
$ % % $ % % $ % $ $ %
$
$ $
3!
$
$ $
$ $
/ &% % # % 4 ## / /% ( ( % % /& / + / &/ / # & & / / &/ #& / & 4 / & & %/ # % / ( & & %&/ % / % / % ( % $/ / 6 0 # # / # & &. . & % .% % /( % ( $ 2% & $
/ / $ / / / + $ & & / + # / # ( &/ & / $ % && / &
33
( %
$
/ /
4 /
$ $ % % % % % $ % % % % % $
$
% $ %
$
%# +
/ / 1 • / & / + # + ( # ( & # - $ - 4 $ / ( / & # - - 4
$
# % & #
/ # % # +- / % # • / # $ # + / % # - # / & # % $ & / & # • # - - 2 & / / / / / 2 / / & # % # / / # &% & $ % $ / &# # + ( & # % && & / # & 2% & # # +
# / & % % # - $%
/ # $ & - % / # % & % $ + # # # / $ & % 1 % & # &% $ & # - $ - # %/ & /% & - $ * # # $ / 2 / / & # & #
3)
$
# & / # /- & $ /
$ $ $ $ $
$ $ $ $
%
/ $ +
#
% # %
1 & & ## / & # ( ( & # % 4 %# / - / / / # / & #% %
# 2% # # / , # + % / % && ( # % & # & & ( & $ % $ / & / $ /
$ $ $ $
$
/ - % / 4 + / # / & / # # / & / % % & - # / , # +# #- / # / / / /& $
%
%
%
%
( 6/
-& / & - & &
( , # & / 1 / ## % / / & / + # , (19.15, 8.37) & , +# / # & #% / / % ( % ## /
$
%
$ $ $ $
$
35 +
/ # & #% ( / // % # %
# # / & #% & & / ') & / % / " & % & #% ( & % / / & # / # & & % / # / / & %% #% & % 2% / / # # &
1 #& / & & 2%% • •
$
% $
% $ $
% !' % $ + # - % # %
# $ q & / q & % q 1 #$ & %& / / ( # / & / / - - - % $ & & % $ ## / / / & / % #& & & & &
#% !
% % % % $ % $ $ % $
,
)
# / %/ 2 & / * + % ( &% + & / / & % / 3 & # // ( +% # /% % % & & ( & %+ / # / 1 & & / & / / / / & % 4 % $ # 2 # • % % # + # & %& / %& / $ & ( #/ ( / &% % %& # & / # -&# / / & •
"
%
# % % &# (
$ &
$
% $ $ $
%
%
%
%
# - & / / # / • $ # / $ $ - # & # •
% $
2 & #
)
# ( , # & # % % % # / / & / +% # / ,&(19.15, / / & 8.37) / % /% # & #% / i/12 & , ( , ) 2% (i − 1)/12 # / # & #% +# # /% # & / # ; ; 9 ; # 99 ;
2 & & / / # &# # # ( +# / ; ; < ;9
/ #
# # - # & / % ( % & & # ( # - " / ' % # / & . 3 &5 & 1 y ,...,y
% % % # " &'
yi ∼ f
'
yi
"
!
%' " ! i = 1, 2, . . . , n
" # $ 1 n " % " $ #" $ " " #
n
#
% $ ( ( # # ) ' '
*
(y1 , . . . , yn )
f
% #
y1 , . . . , yn ∼
f (y1 , . . . , yn ) =
' "! ' ' '# '
Qn
i=1
f (yi)
'
f
' '# ' % # % " %'! !%'!% %' '!% & ' " # !' ' + # ( % % % , % # % - # $ %$ . % ) $ "% - ) # $ "% / $ ' $ # %( %% # " % # # ( " . # ( , %( ) 0# $ ' ( &1 1 2 - % , ) # % " # $ %$ % - % # # # %( %% " 3 4 %( % # 5 ( 6 ( 4( ' R R 4( % ' '# ' ' 7 # 2 y1 , y2 , . . . , ∼ f µ = yf (y) dy σ = (y − $ # " ' * # ( / - 2
µ) f (y) dy
f
µ
σ
2
# - / %$ # ( $ ' - " ( % ) $ " % % % ) % $ ' y¯n ≡ (y1 + · · · + yn )/n n µ 4 $ ( % % % #
•
0 %$ " %
•
•
%
y¯n
n
%
- $ ( %
n
) - ( %
µ
%
n
%%
y¯n
% % $ "
µ
# # % # " (
* $ % & '& # & '& # %- % % ( % % ' " % $ - # %$ $ %( % ( $ # " '
!%
$
!
! %'
x1 , . . . , xn E[x1 + · · · + xn ] = µ1 + · · · + µn
%( % %
E[x1 + x2 ] = =
Z Z
ZZ
n=2
!"
!%"
y¯n
µ1 , . . . , µn
'
(x1 + x2 )f (x1 , x2 ) dx1 dx2 ZZ x1 f (x1 , x2 ) dx1 dx2 + x2 f (x1 , x2 ) dx1 dx2
= µ1 + µ2
!! !%
!
y1 , . . . , yn
! %'
" ! $
$ * " - % " $ * $ % & ' # & ' '
f
!%
E[¯ yn ] = µ
! ! %'! !%'!% %' x1 , . . . , xn %' " !%
!"
µ
!%"
µ1 , . . . , µn σ1 , . . . , σn Var[x1 + · · · + xn ] = σ12 + · · · + σn2 $ %( % % ' 3 % * $ &' ) n=2 Var(X1 + X2 ) = E((X1 + X2 )2 ) − (µ1 + µ2 )2 = E(X12 ) + 2E(X1 X2 ) + E(X22 ) − µ21 − 2µ1 µ2 − µ22 = E(X12 ) − µ21 + E(X22 ) − µ22 + 2 (E(X1 X2 ) − µ1 µ2 ) = σ12 + σ22 + 2 (E(X1 X2 ) − µ1 µ2 ) .
(
"
X1 ⊥ X2
2
E(X1 X2 ) =
ZZ
x1 x2 f (x1 , x2 ) dx1 dx2 Z = x1 x2 f (x2 ) dx2 f (x1 ) dx1 Z = µ2 x1 f (x1 ) dx1 = µ1 µ2 . Z
4
Var(X1 + X2 ) = σ12 + σ22
'
* $ &'1 ( % # # - * $ & ' # %
'
!!
y1 , . . . , yn f 2 2
σ
Var(¯ yn ) = σ /n
* " - % " $ * $ % & ' # &'1 '
! " # $ %& '( ) * ! +
µ
, σ > 0-
σ = = ≥
P[|X − µ| ≥ ] ≤ σ 2 /2 .
2
X
Z
Z
(x − µ)2 f (x) dx µ− 2
−∞ µ−
Z
−∞
2
≥
Z
(x − µ) f (x) dx + (x − µ)2 f (x) dx +
µ−
−∞
2
f (x) dx +
= 2 P[|X − µ| ≥ ].
Z
Z
µ+ 2
µ− ∞
Z
∞
µ+
(x − µ) f (x) dx +
Z
∞
µ+
(x − µ)2 f (x) dx
(x − µ)2 f (x) dx
f (x) dx
µ+
* $ % &'& # & '& - $ $ $ % " % % % ' * # %- % ) % % ) ( % % 2 '
2
!
* " &+
y1 , . . . , yn
2 µ σ , > 0-
lim P[|¯ yn − µ| < ] = 1.
n→∞
5 % ( * $ & '& &
y¯n
& '&2
'
lim P[|¯ yn − µ| < ] = lim 1 − P[|¯ yn − µ| ≥ ] ≥ lim 1 − σ 2 /n2 = 1.
n→∞
n→∞
%
n→∞
" * $ &'& % # 4 7 - " 7 ( $ .
% '
! (
* " & +
y1 , . . . , yn
2 µ σ , > 0-
P[ lim |Y¯n − µ| < ] = 1; n→∞
-
P[ lim Y¯n = µ] = 1. n→∞
% # % " % % , # - 7 7 # 47 7 ' 4 4 '1 ' ! ( ! + y , . . . , yn √
2 1 f µ σ zn = (¯ yn − µ)/(σ/ n) - , a < b
lim P[zn ∈ [a, b]] =
n→∞
Z
b a
1 2 √ e−w /2 dw. 2π
z (0, 1) n * 7 - " 7 ( $ % % - $ /% % $ ( % - / # -
%$ % %$ ' % % % ( $ " % $ ( - % % %$ % - % ) ) " % $ ( % n→∞ " %$ % % # % ( (
X¯ n →
'
2
0 % ) , $ & '&& ) - - ( %# % $ ( % $
µ 4 - % 5 % ) % $ $ $ # $ ( % P[ ] ( $ " % $ ( % %# " $ ) ) # & ' * 5 7 $ * $ % ( % / % % $ ( % $ %' 5 ( ) 5 7 $ * $ % %
lim pzn =
n→∞
% % ) "
pz n ≈
- ( $ %
py¯n ≈
(0, 1). n
)
(0, 1),
√ (µ, σ/ n).
, $ & '& & - % $ ( # $ " 5 % " ) ) * % % $ ( % # # 0 ( & ' ' * ( % - % % $ " & % $ ( % ) ( % ' 0 % = 50 % $ ( - ( $ % " % % $ . = 50 X1 , . . . , X50 √ ( % ' ) % #
' Xi ∼ (.493) µ = .493 σ = .493 ∗ .507 ≈ .5 * " ) # 5 7 $ * $ ) - # & '
= 50
√ ¯ n ∼ (µ, σ/ n) ≈ (.493, .071) X
* % % $ # % # ( " 0 ( &' ' % # , $ % $ ' # % (.493, .071) % ( % ¯ 50 ∼ (.493, .071) ) , $ ' * 5 7 $ θˆ = X * $ % % , $ - # " ' % % n % ( ' 4 '& ' - # %( %% ( % " - % n = 50 n ( ' 4 $ )
¯ n ∼ (.493, .035) X ¯ n ∼ (.493, .016) X
- -
= 200 = 1000.
* % # % % # $ # # # - % " 0 ( & ' ' * 5 7 $ * $ $ /% % $ % ( # % (. " %$ %
y¯n zn
&'
E[¯ yn ] = µ E[zn ] = 0
)
0 1 2 3 4 5 6
Density
2
0.3
0.4
0.5
0.6
0.7
0.6
0.7
0.6
0.7
6 4 0
2
Density
8 10
theta hat n.sim = 50
0.3
0.4
0.5
10 0
5
Density
20
theta hat n.sim = 200
0.3
0.4
0.5 theta hat n.sim = 1000
0 ( & ' % $ % " % % $ ( % ' 4 # ( % $ . , $ % # 5 7 $ * $ '
'
'
2
√ ) # SD(¯ yn ) = σ/ n SD(zn ) = 1 y¯n zn
% ) , $ ) $ # % ( '
* % - " % # / - " $ * $ % & ' # &'1 ' % % # % / 5 7 $ * $ ' %( $ "$ 5 7 $ * $ % # % ( % " y¯n # %$ % # $ # % #
" ( % zn µ σ " $ '
f
% $ %( " & ' 4 - " µ # % % %
A1 , . . . , An
µ(
n [
i=1
'
* !
( # !! &
% $ (
Ai ) =
n X
n ≥ 2
) #
µ(Ai ).
i=1
# % ' 5 ( ( $ " & % ) % ) ' ' ' )
( , ( & " ( $ ' - ( , #
% '
- % - % ( %(
- " - ( # ( , $ & & % ( % $ ( % $ %- '
(
& # $ % $ % # ( % # ( - - # ( ( % $ " $ % ' * ( ( ) $ ( % / " $ # ( ' " ' % $ % ( ) % $ / # ( A≥2 A D / % $ # - "$ & $ % ' 4 % & $ $ ( % " # / ( ) $ % / - $ $ ( $ " & # $ , $ ( $ " $ % ' " min(3, A − 1)
% / % $ % ( ) $ # " # $ %" D≥1 D ( % $ $ ( $ " & # $ , $ ( $ " $ % ' % min(2, D) $ % - % % / # # " # - $ , $ ( $ $ %%
( $ " $ % '
'
22
/ % - $ % % % # ' a a # " # % - $ % % # ' % % # % $ #
% % ' d" % ( %d % - # ) % % #
% % % $ # % % # % ' 0 $ % )
" % # % % - % # $ % $ "$
# - % - % # $ % $ " $ # '
- $ % % ) " - $ % $ # " $ # '
• $ / - #
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% # $ ' % ) '
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[0, 1] p(u) = 1
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V = U2 V V % "( " ' ' 0 # # " " ' 6 U V pV (v) v
W = 2U W W % "( " ' ' 0 # # " " ' 6 U W pW (w) w
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7
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# - # % " ' X 0 # # % " ' Y $ # % ( " ' Y # $ % % % ( #
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0 # ' p(x, y) # # # X Y # 0 # $ # % % ' p(x) p(y) " # # 0 # # # % % ' p(x | y) p(y | x) 0 # ) ) # ' E[X] E[X | Y = .5] E[X | Y = −.5] $ , $ & ' & ' " ! ' argmaxλ P[x = 3 | λ] = 3 # ( & ' ' & ' % #" " ( ( % # $ ' 0 # R ' p w R p(s) ds 0 # R " # " ( &' ' R p(s) ds # & ' 6 2 % ! % ! #" $ ( % % %" ! - % ' ' ' " ' " $ % " - ! % ( ' , p(y) ≥ 0 " y & '
%
( ( % # $ ' 0 # R 0 ' y −∞ p(s) ds ( % # % % % $ ( $ & ' % ( ) ' y ∈ (0, 1] % p(y) % ' " $ & ' 4 $ , % % # % % " " $ # ' % ( # ( % # " % ( ( " #"
"
1 2 1 p(y) = √ e− 2 y 2π
1&
( % % $ / 6 6 " ' X ∼ (λ) E(X) = λ - ( % # # $ " $ # % ( ' 6 " % ' X ' ∼ (µ, σ) E(X) = µ " " 6 " (% &1 '
' * $ &X'1 ' ∼ (n, p) Var(X) = np(1 − p) 6 " 6 ( % % $ ' / - ( %X # ∼ # (λ) $ Var(X) " = λ $ # % ( # * $ & ' ' , ) # " ' ( % * $ & ' ' (λ) Var(X) X∼ # " ) # ' ( % * $ & ' ' X ∼ (µ, σ) Var(X) ' ( % " ! % " ( & '& ' ( % " ! % " ( & '& ' ! % ' " ) # & ' 7
' X1 ∼ (50, .1) X2 ∼ (50, .9) X1 ⊥ X2 Y = # % ( ! - ! ' % (100, .5) X1 + X2
Y # ) ' 7
' X1 ∼ (50, .5) X2 ∼ (50, .5) X1 ⊥ X2 Y1 = # ' % $ X1 + X2 Y2 = 2X1 "" Y1 Y2 % ( % " ! ! ( %- ' "" Y1 Y2 % ( ' $ % ' 5 % # ( % $ % " % , $ ! , - " # % ( . $ # # % # % % ( ( ' λ ' ') $ % # . −λt ' %%( $ %( %% p(t) = λe # " ' 7 $ " , ( % $ # # # $T1 ( " % # ( % $ ' T2 % # % ! " (T1 , T2 ) 7 ) $ ( , - ( % $ % ' S =% T1 + T2 ' ' ! % ( % $ % P[S ≤ 5] - , $ ( % % E(S) $ ! % ( " - ( % # ' " % " - ' * % % # ( - " ( # ' " " % # % ( &2 '
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)
1
% ' ' 6 * $ & ' ' & # % " % ' & ' 6 * $ & ' ' 2 ( ( % " # ' % # $ ! % % $ .# ( % ' 7 " xi yi % " $ # # ( ) % ! ' * ( i " " 0 % # # # xi xj # # # xi yj # # # yi yj # # # # xi yi ' % " % $ $ - / % # - # # # %( % ' 7 & %( " % # # %( " % # ' % ! & & & "
% $ # # 4 ) # %. ' # %( % %
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i n−1
≡ y(i+1)
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qp " p q .5 (q .25 , q.5 , q.75 ) " q.1 , q.2 , . . . q.78 # i i+1 p ∈ ( n−1 , n−1 )
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0.4 0.0
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y
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n " 2 s ( & ) * & ( & q.75 − q.25 "
"
! " " " ! " " ! f " ! " f " ! "
f " " f " " " ! " " ! " ! " "
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3
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5
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5 10
20
30
0
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30
OJ, 1
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0.0 0.5 1.0 1.5 2.0
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! ! " " "
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1.0 0.0
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0 10
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0.0
0
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0.5
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2
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1.5 0.0
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2
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X ˆ dx = 1 f(x) n i −∞
Z
∞
−∞
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(b)
0
0
1
2
2
4
3
6
4
5
8
(a)
140
160
180
120
140
160
calories
calories
(c)
(d)
180
0
0.010 0.000
1
2
3
density
4
5
120
120 140 160 180
100
calories
(e)
(f)
200
0.000
0.010
0.015
density
0.020
0.030
calories
0.000
density
100
140 calories
150
180
100
140
180
calories
! + !" + !# + " " " ! + !& + ! + ! "
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! ! %
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% %
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%
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1 0.5
dose
2
(a)
5
10
15
20
25
30
35
25
30
35
25
30
35
growth
1 0.5
dose
2
(b)
5
10
15
20 growth
OJ
method
VC
(c)
5
10
15
20 growth
! ! " " " " ! ! " " "
% & ! " ! !
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" ! " " " " " " " n = 213 x , . . . , x 1 213 ! ! ! " (x , . . . , x(213) ) " (1) E[(z(1) , . . . , z(213) )] ! ! x i " ! " " ! !
0
2
4
6
8
10
Individual quizzes
Q1 Q4 Q7
Q11
Q15
Q19
Q23
Student averages
0
2
4
6
8
10
score
"
$
! " !"
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%
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latitude = 45 longitude = −30
latitude = 45 longitude = −20
4
9 7
4.5
6
5
6
5.5
8
7
8
6.5
9
10
latitude = 45 longitude = −40
−3 −1
1
3
−2
0
2
−2
0
2
n = 105
n = 112
latitude = 35 longitude = −40
latitude = 35 longitude = −30
latitude = 35 longitude = −20
5.5
8.5
8.0
6.5
8.5
9.5
9.0
7.5
9.5
10.5
n = 213
−2
0
2
−2
0 1 2
−2
0
2
n = 24
n = 44
latitude = 25 longitude = −40
latitude = 25 longitude = −30
latitude = 25 longitude = −20
−2
0
2
7.2 −2
n = 47
" "
6.8
6.5
6.5
7.0
7.5
7.5
8.5
8.0
7.6
n = 37
0
n = 35
2
−2
◦
0
n = 27
2
%
" '
" ! !
% !
" % % ! ! % !
% % ! & ! % & % "
!
! !
1198/(1198 + 1493) = 44.5% 557/(557 + 1493) = 30.4%
!
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! % ! ! ! ! ! " !
% "
! % (
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%
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Student admissions at UC Berkeley Rejected
Female
Gender
Male
Admitted
Admit
$ "
Student admissions at UC Berkeley Female
Admitted
Male
Rejected
Admit
Gender
"
% ! #
% ! ! !
" " ! " % % %
! !
!
!
! % " % ! " ' % ! %
! " % & ! * + $ '
%
%
$
! %
!
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% ! ! !
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! % % %
&
! ! & " % " ! !
!
(a)
2
3
4
$
(c)
5
6
40
60
80
duration (min)
waiting (min)
(b)
(d)
100
0.01
0.02
Density
0.3 0.2 0.0
0.00
0.1
Density
0.4
0.03
0.5
1
1
2
3
4
5
duration (min)
6
40
60
80
100
waiting (min)
! " "
50
60
70
80
90
time to next eruption
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
duration of eruption
"
#
"
1.5
2.5
3.5
duration
4.5
a
0
50
100
150
200
250
200
250
data number
50
70
waiting time
90
b
0
50
100
150
data number
"
" "
3.5 1.5
duration
a1
0
50
100
150
data number
4.0 2.0
duration
a2
0
20
40
60
80
100
120
140
data number
80 50
waiting time
b1
0
50
100
150
data number
80 50
waiting time
b2
0
20
40
60
80
100
120
140
data number
" ! " "
" ! " " " " " " " "
" " " " " −10) " " (−17,
! " ! ! ! ! 35◦
" " " ! " " ! (−20, −13) (−25, −16) (−30, −20)
(−34, −25) (−40, −28) !
" " ! # " • ! • " " #
!
" " " " ! ! " " "
" " ! # " ! " " " •
"
!
! ! "
X∼ # ! (λ) θ) λ X ∼ ! (n, θ ! X X ∼ (µ, σ)
! ! µ σ ! # (µ, σ) λ !λ θ θ (µ, σ) , ! " " ! " " ! " ! ! ! ! " " $ " !
Given : lon −40
30
−30
40
−25
−20
50
20
−15
30
−10
40
50
5
10
15
temp
5
10
15
20
−35
20
30
40
50
lat
"
"
Given : lat 20
−30
30
−20
35
40
−10
−40
45
−30
50
−20
−10
5
10
15
−40
25
5
10
15
temp
−40
−30
−20
−10
lon
"
"
" X∼ (λ) ! λ # " X ! ! λ Pr[X = 3 | λ] ! λ `(λ) " " X=3 λ ! ! ! ! θ X , . . . , Xn ∼ f (x | θ) 1 θ Y f (X1 , . . . , Xn | θ) = f (Xi | θ). # θ f( " ! | θ) θ "
`(θ) ) `(θ | ! `(θ 1 θ2 )/`(θ2 ) > 1 θ1 θ
k ! θ 2 )/`(θ ) = k `(θ
1 2 1 X
X
NR, R, NR, NR, NR, R, NR, NR, NR, R
X = 3 3 7 X ∼ (10, θ) `A (θ) = 10 θ (1 − θ) ! 3 " ! ! ! " " ! θ θ ≈ 0.3 θ ! θ ≈ 0.3 ! ! θ ≈ 0.1 θ ≈ 0.6
θ " θ ≈ 0.3
"
0.8 0.6 0.4 0.2 0.0
likelihood function
1.0
0.0
0.2
0.4
0.6
0.8
1.0
θ
"
`(θ)
θ
$
! "
"
•
!
!
" " !
! ! Y Y = 10 `B (θ) = P[Y = 10 | θ] = P[ " 9 2 θ (1 − θ)7 × θ = 2 9 3 θ (1 − θ)7 = 2
$
] × P[ ]
! 9 10 `A ! 2 / 3 ! θ
" ! ! Z Zk 1 k # Zi " ! i ! !
`B
`C (θ) = (1 − θ)θ(1 − θ)(1 − θ)(1 − θ)θ(1 − θ)(1 − θ)(1 − θ)θ = θ3 (1 − θ)7
!
! `A `B ! " ! "
θ ! " % & && +
`C
! % ! " ! ! " & ! ' !
`(λ) ≡ p( | λ) = p(y1 , . . . , y60 | λ) =
60 Y 1
=
Q
p(yi | λ)
60 Y e−λ λyi
yi ! 1 −60λ 40
∝e
λ
" !
yi ! λ " P " `(λ) `(λ) y i ' " ' P yi yi = 40 # yi `(λ) ' % ! ! ! " !
λ
0.4
0.8
!
0.0
likelihood
0.0
0.5
1.0
1.5
2.0
λ
"
"
`(θ)
P
yi = 40
% & &
"
! !
# ! % ! "
!
! # ! ! "
! " " ! ! ! & " % % ! !
# ! " " % ! ! ! !
" ! % ! " %
!
x=8
%
X
% ! " "
%
' !
X∼
(145, θ)
% "
`(θ) ∝ θ8 (1 − θ)137
" ! θ % "
likelihood
% " ! ! % % θ %
0.00
0.05
0.10
0.15
0.20
θ
"
$
% !
!
& ! ! " ! θ
`(θ)
' ! '
! ! ! ! ! ! " " " ! " ! " " ! "
" " " ! #
! n P[n | ] = 2−n P[n | ] = n 1 "
2
" " ≈ 1/8 argmax `(θ) ≈ .055 `(.025)/`(.055) ≈ .13
θ = .025 θ = .055 " "
" !
" θ = .1 `(.011)/`(.055) ≈ `(.15)/`(.055) ≈ .001 θ = .011 θ = .15 "
" θ θ = .025 θ = .1 θ ≈ .055 ! θ " θ
!!
! " ! " ! ! # ! f X1 , X2 ,. .. , Xn f
" f µ σ (µ, σ) ! µ = Q `(µ, σ) f ( | µ, σ) = f (Xi | µ, σ) f `(µ, σ) " µ ¯ ¯ X µ δi ≡ Xi − X ! ¯ X, i = 1, . . . , n n " √ ¯ ∼ (µ, σ/ n), ! X
#
2 ! σ σ ˆ 2 = s2 =
!
X
1 `M (µ) ∝ exp − 2
δi2 /n
¯ µ−X √ σ ˆ/ n
2 !
!
! $$ !
" "
# " "
"
!
% & & &
' '
X i ! ¯ = 8/145 ≈ .055 σˆ 2 = (8(137/145)2 + X 2
137(8/145) ))/145 ≈ .052
σ ˆ ≈ .23
1 `M (µ) ∝ exp − 2
µ − .055 √ .23/ 145
2 !
' !
marginal exact
0.0
0.4
likelihood
0.8
' % '
0.00
0.05
0.10
0.15
0.20
µ
"
"
% !
!
% &
# % % " %
! ! ! %
% % % &
& ! & "
' % & " "
" ! ! " ! & ' & % ! $ !
& ! ' ' (
% !
! # ! & " ' !
` (µ)
n = 60 µ
M
! " ! % ! " % # ¯ % ! ' X ' !
`M (µ)
!
(a)
0
200
400
600
800
1000
Salary (thousands of dollars)
0.2
0.6
1.0
(b)
likelihood
340
360
380
400
420
440
460
mean salary
"
"
!
!
"
% !
•
• •
•
!
%
% %
!
! % !
% %
!
# Xi (µ, σ) (µ, σ) % & &
% # '
# 2
!
#
$
%
! & % ! % " % %
" ! %
% " % ! " ! % % % % " ! (µ, σ) %
' !
`(µ, σ) = =
n Y
1 n Y 1
f (xi | µ, σ) √
1 2 1 e− 2σ2 (xi −µ) 2πσ 1
∝ σ −n e− 2σ2
Pn
2 1 (xi −µ)
% ! ' ! ' !
(µ, σ) ≈ (1.27, .098)
" ! % (µ, σ) ' % !
!
" ! (µ, σ) %
µ
!
# " 2 % σ ! ! % % " ` µ σ % % ! " ! µ
' ! ! σ = .09, .10, .11 % " !
µ
" ! " ! " µ & " ! ! ! "
σ
' ! !
µ
' " ! " ! µ µ % " " "
(1.25, 1.28)
(b)
0
0.10 0.08
1
0.09
2
σ
3
0.11
4
5
0.12
(a)
1.0
1.2
1.4
1.6
basal area ratio
1.20
1.30 µ
0.6 0.4 0.0
0.2
likelihood
0.8
1.0
(c)
1.20
1.30 µ
$$" "$$ " ÷
•
%
'
•
! " '
•
" !
" "
"
`(µ, σ)
µ
σ
"
!
"
•
" " '
•
&
" " ! " "
% '
•
!
$
"
%
•
' !
! " !
σ
% &
& &
σ = .09, .10, .11
%
% % % !
! !
" " % % % % %
(µ, σ)
' ! " ! ! µ % % " ! ! % % σ ! ! ! % (µ, σ) " ! &
'
!
%
0.8
0.9
1.0
1.1
1.2
1.3
0.7
σ
6.8
7.0
7.2
7.4
7.6
µ
"
!
! " ! " ! ! " ! " ! ! % & && &
% " %
% " "
! ' " " ' ! % &
" " ' % " % !
"
t
" %
t−1
% !
# !
! ! ! ! % % !
& % " %
" !
T % % ! N i ! % i NiO % " % ! i T
Ni ∼
(λ)
$$ $$
θf
"
"
$$ $$ "
! % !
θf
! % ! % ! ! ! ! & "
" T θf NO ∼ (NiT , θf ) NiO Ni % " i % (λ, θf ) ! ' O
N1992 = 0
% ' !
`(λ, θf ) = P[N O = 0 | λ, θf ] ∞ X = P[N O = 0, N T = n | λ, θf ] n=0
= =
∞ X
n=0 ∞ X n=0
=
P[N T = n | λ] P[N O = 0 | N T , θf ]
e−λ λn (1 − θf )n n!
∞ X e−λ(1−θf ) (λ(1 − θf ))n n=0 −λθf
eλθf n!
=e
! ! " log `(λ, θf ) log ` `
10
10 ! " ! %
% (λ, θf ) = (2.5, 1) (λ, θf ) = " % O N =0
log10 `(λ, θf ) = −1 (λ, θf ) (6, .4)
λ=0
θ =0
f % ! ! " % !
!
λ
" & " !
λ
" % ' % ! % "
θf
% " !
% % " ! λ " " ! " % ' !
λ
% " ! " ! " !
θf θf
! %
" % ' !
λ
' " ! ! " ! λ λ % % " !
%
&
θf
% ' ! !
! !
λ
θ
' %
f"
!
•
%
!
0.4 0.0
θf
0.8
(a)
0
1
2
3
4
5
6
λ
0.0
0.4
0.8
(b)
θf
0
200
400
600
800
1000
λ
"
!
"
(λ, θf )
"
#
! ! ! " ! ! " θ ! ! " t t ∼ (θ, σt ) ! σ t
! " θ `M (θ)
! "
! y ! y θ " p(y | θ)
! `(θ) θ θ θ `(θ) `(θ1 ) > `(θ2 ) θ1 θ2 ! " θ1 θ2 `(θ) `(θ1 ) = 2`(θ2 ) θ1 θ2 "
θ θˆ ! θ θ y
`(θ) " !
! θ `(θ) θ ˆ θ
θˆ ≡ argmaxθ p(y | θ) = argmaxθ `(θ).
$
" θ `(θ) y = 8 ! $ ˆ θ ≈ .05. " ""
$
#
! `(θ) " " `(θ) ∝ θ8(1 − θ)137
"
!
`(θ) ∝ 8θ7 (1 − θ)137 − 137θ8 (1 − θ)136 θ = θ7 (1 − θ)136 [8(1 − θ) − 137θ]
0 = 8(1 − θ) − 137θ 145θ = 8 θ = 8/145 ≈ .055
θˆ ≈ .055 `
! ! " " ! ˆ y y " θ = y/n n − ! " log `(θ) = argmax log(`(θ)) Q ! argmax log ` `(θ) = p(y | θ) log `(θ) = " P i θ) log p(x i |
log `(θ) = 8 log θ + 137 log(1 − θ) log `(θ) 8 137 = − θ θ 1−θ 137 8 = 1−θ θ 137θ = 8 − 8θ 8 . θ= 145
y1 , . . . , yn ∼ (θ) X
yi = θˆ = n−1 ! "
θ
y1 , . . . , yn ∼
(µ, σ)
µ ˆ = n−1
y1 , . . . , yn ∼
ˆ = n−1 λ
y1 , . . . , yn ∼
(λ)
X
yi =
X
λ
µ
yi =
ˆ = n−1 λ
X
(λ)
λ
yi =
! ! " ˆ " θ θ ˆ ! ! θ θ " " !
! ! ˆ " " θ `(θ) > `(θ)/10 ! "
.1
≡
(
`(θ) θ: ≥ .1 ˆ `(θ)
)
! ! " α ∈ (0, 1) α
α
≡
(
`(θ) θ: ≥α ˆ `(θ)
)
! θ
α ! θ ! ! α ! ! # α θ !
α ≈ .1
! `(θ) ! ! ˆ " θ " θ → " $ ±∞ ˆ " θ ≈ θˆ ⇒ `(θ) ≈ `(θ) ˆ θˆ θ θ α
θl θu
α
= [θl , θu ]
!
" ˆ ˆ `(θ) θ = 8/145 ! "
! ! !! θl θu ≈ .013 ≈ .015 .1 ≈ [.023, .105] $ " θ " " θ " " θ ! ! ! ! θ ! ! " " # n → ∞ "" " θ
% &
% & % %
!
θ
!
! " ! % ! ' !
`(θ)
`(θ)
" !
" % ' !
!
!
!
≈ [.15, .95] `(θ) !
.1
! !
≈ [.05, .55] [.25, .7]
' ! .1 %
.1 ' " ! %
[.38, .61]
0.6 0.4 0.0
0.2
likelihood
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
θ
"
! θ " " "
$
!
! " " ! ! ! !
ˆ θ y1 , . . . , yn ˆ ˆ " " θ θ ˆ #
Fθˆ θ
! Fθˆ " " n = 50, 200, 1000 θ
! $ θˆ " Fθˆ θˆ ˆ θ "
! F ˆ "
! θ ˆ " θ θ Fθˆ ! ˆ ! ! Fθˆ " θ θ "
!
! ˆ " θ θ !
ˆ ˆ θ1 θ2 Fθˆ1 Fθˆ2 ! " " θ y1 , . . . , yn FY # θP≡ E[Y ] ˆ θ1 = (1/n) yi θˆ2 " n = 4, 16, 64, 256
ˆ FY (0, 1) θ1 ˆ θ2 " " " F Y # F Y " ˆ ˆ θ2 " "" " ˆ !θ1
" θ1 θ ˆ " θ2
! "
"
ˆ ˆ ! ! " θ1 θ2 ˆ θ
(b)
−2
−1.0
−1
0
0.0
1
0.5
(a)
mean
median
mean
(d)
−0.4
−0.2
0.0
0.0
0.4
0.2
(c)
median
mean
median
mean
median
ˆ " " θ θˆ2 1
! !
! " ! ! # FY
ˆ ! ! ! " θ !
Fθˆ
θˆ ∼
(µθˆ, σθˆ).
µθˆ = µY σY σθˆ = √ n
"
!
" " ! # n = 5, 10, 25, 100 p = .1 " ! !
"
! "
"
0.2
0.4
0.6
0.0
0.4
(c)
(d)
0.6
0.0
0.2
0.4
0.6
θ^
ˆ " " !θ
! !
0 2 4 6 8
density
12
θ^
0 2 4 6 8
density
0.2
θ^
12
0.0
0 2 4 6 8
density
0 2 4 6 8
density
12
(b)
12
(a)
0.0
0.2
0.4
0.6
θ^
(n, .1)
"
! ! " n ! " FY " " n = 5 n " ! n " " " ! n ! 256 n = ˆ
θ " θˆ "
"
!
ˆ # ˆ ! ! |θ − θ| θ ∼ (µθˆ, σθˆ) σ θˆ $
±2
Pr[|θˆ − θ| ≤ 2σθˆ] ≈ .95
$
$ ! # !
! " ! " `(θ) " Fθˆ " √ # ˆ θ ∼ (θ, σθˆ) σθˆ ≈ σ/ n ! θ $ ±2σθ (θˆ − 2σθˆ, θˆ + 2σθˆ) #
" " 2 ¯ ˆ ˆ √Y `(θ) ≈ exp − 21 σˆθ− ! .1 ≈ (θ −2σθˆ, θ +2σθˆ) / n
# θ! p(θ) p(θ) " " θ p(θ) θ p(θ) ! # $ θ " & p(θ) ! % ! # " & & $ !
( ! '
! ! P( ) = P(' ) = 1/2 ! ) P( ) = 1 P(' ) = 1
' ! # ! # ! *+ * , * , *+
& −$1 + P[ ! ] × $10 !
* +
! !
• • • • • •
P[
] ≥ .1 !
!
#
" " !
#
!
#
!
" " !
" " !
! θ " "" !
• #
! " ! " θ
p(θ) θ! ' " p(y1 , . . . , yn | θ) y1 , . . . , yn θ! " p(y , . . . , yn , θ) ! 1 " p(θ | y1, . . . , yn ) θ y1 , . . . , yn ! # p(θ | y1 , . . . , yn ) ! p(θ) p(θ | y1, . . . , yn ) !
* *
# ! & * *+ ++ !
( $ ' ! " =1 " ! ' ' =1 =0 " " ' =0 $ !# P[' = 1 | = 1] = P[' = 0 | = 0] = .95 ! " "
P[ = 1 | ' = 1] # " "
( , ') ' ! ' "
'! ' P[ = 1 ' = 1] P[ = 1 | ' = 1] = P[' = 1] P[ = 1 ' = 1] = P[' = 1 = 1] + P[' = 1 = 0]
P[ = 1] P[' = 1 | = 1] = P[ = 1] P[' = 1 | = 1] + P[ = 0] P[' = 1 | = 0] (.001)(.95) = (.001)(.95) + (.999)(.05) .00095 = ≈ .019. .00095 + .04995 $ !
" " ' " $
! *+ + + * ! " " ( $$$ " + " ( * " *
$ ! ' ! ' ! # ' "
*
=0
= 1!
`(0) = .05;
`(1) = .95
$ " ! P[ = 1 | ' = 1] ' = 1] P[ = 1 = ' = 1] P[ = 0 | ' = 1] P[ = 0
P[ = 1] P[' = 1 | = 1] = P[ = 0] P[' = 1 | = 0] P[' = 1 | = 1] P[ = 1] = P[ = 0] P[' = 1 | = 0] 1 .95 = 999 .05 ≈ .019
"
' ! ' ! " ! # ! $ " " ! y y1 , . . . , yn !
p(θ, y) p(y) p(θ, y) =R p(θ, y) dθ p(θ)p(y | θ) =R p(θ)p(y | θ) dθ
p(θ | y) =
!*+ !$
"
p(θ | y) = R
!*+
!
p(θ | y)
p(θ)`(θ) p(θ)`(θ) dθ
θ
*
p(θ | y) =
p(θ)`(θ) c
R
c = p(θ)`(θ) dθ θ! # ( θ & θ θ! ' c * R p(θ | y) dθ = 1 ! # c ! ! ! c! * * p(θ | y) ∝ p(θ)`(θ) !
#
! *+ c! !* * ! *!
cR c = [ p(θ)`(θ) dθ]−1 ! !
% ! " (λ) % % ! " !
%
p(λ) = 4λ2 e−2λ (3, 1/2) ! p(y | λ) ∝ λ3 e−λ ! ! p(λ | y) ∝ λ5 e−3λ ! ! (6, 1/3) c % ( c = 6 1/[5! × (1/3) ]
! "
! $ " " ! !
y1 , . . . , yn
`(λ) =
Y
p(yi | λ) =
Y e−λ λyi yi !
∝ e−nλ λ
P
yi
!
0.5
*
0.0
0.1
0.2
0.3
0.4
prior likelihood posterior
0
1
2
3
4
5
λ
" ! " y=3
λ
*
$ ' ! * ! ! `(λ) n = 1, 4, 16 y¯ = 3 ! n = 1, 4, 16 * ! # n !
$ n λ '
λ ! ' " λ " !
!
# n
$ `(λ) ! ' n ! " ! ! ! log `(λ) = c + # P n log p(λ) + 1 log p(yi | λ) ! # n → ∞ " log p(λ) ! $ # * y¯ + n+ ! ! ! ! # * ! p(θ | y1, . . . , y60 ) ∝ λ42 e−62λ
+ " (43, 1/62) ! ! ! ! !!* ! " * $$$
!
! ! " ! " "
*
b
0.6 0.0
0.0
0.1
0.2
0.4
likelihood
0.3 0.2
prior
0.4
0.8
0.5
a
0
1
2
3
4
5
0
λ
1
2
3
4
5
λ
0.6 0.4
n=1 n=4 n = 16
0.0
0.2
posterior
0.8
1.0
c
0
1
2
3
4
5
λ
$ !
1, 4, 16
λ
n =
*
prior likelihood posterior
2 0
1
density
3
0.0
0.5
1.0
1.5
2.0
λ
+ P
!
yi = 40 !
λ
n = 60
* "
! & % ! ! &
"
" "
! "
% !
% %
θ
% %
" ! ! " % ! " ! " ! ! ! # 4.2/145 ≈ .03 " ! % " ! % !
" ! θ ≈ .03 " % % ! ! % !
% ! θ ≈ .06 ! Γ(20)Γ(400) 19 θ (1 − θ)399 p(θ) = Γ(420)
!*
! (20, 400) ! `(θ) ∝ θ8 (1−θ)137 p(θ | y) ∝ θ27 (1−θ)536 (28, 537)
!
p(θ | y) =
Γ(28)Γ(537) 27 θ (1 − θ)536 Γ(565)
* * " " ! ! !
a −bλ a λ e θ (1 − θ)b
c
" ! ' ! ' " ! ' " " !
* $
40
10
20
30
prior likelihood posterior
0
0.00
0.05
0.10
0.15
0.20
θ
* !
*" +
" # " y1 , . . . , yn , yf p(· | θ) ! y1 , . . . , yn !
yf yf ! ' ! yf ! # !
" # " " " " y !# f " ! ' (0, 5) " $ + " yf Pr[y ∈ (0, 5)] = .90 ! " f " " $ + " $ " + " ! " " ! # $ + " " $ ! (0, 5) $ + " " (−1, 4.5) !
" # " & yf ! " ! $ $ θ ! # " y1 , . . . , yn ! $ $ θ! y1 , . . . , yn , yf ∼ ! ! ! (−2, 1) !
θ ' $
y1 , . . . , yn ! ' yf θ! y1 , . . . , yn yf " θ!
p(yf | θ, y1, . . . , yn ) = p(yf | θ).
' y1 , . . . , yn !
θ
*" *
# yf yˆf = −2 $ + ! (−2, 1) " # (−∞, −0.72) (−3.65, −0.36) (−3.28, ∞) !
" " " ! # yf " yf (−2, 1) ! yf ! $
θ! ' ! . . , yn θ y1 , . ˆ p(y | θ) ! f " y ,...,y ! 1 n µ ˆ = −2 σ ˆ =1 " ! * * !! ! " #
ˆ λ λ = 2/3 ! ' ! " ! ! yˆf = 0
$ " {0, 1, 2} ! ' $$ " ( ≈ .97 {0, 1, 2, 3} ! !
0.4 0.2 0.0
probability
0
!
2
4
6
"
8
yf ∼
10
(λ = 2/3)
*"
' $ " ! " θ ! # yf $ θ ! $ ! ' ! " y1 , . . . , yn θ "
! θ ! " ! " " " ! $ " " θ! #
! ' θ " " y1 , . . . , yn , yf θ y1 , . . . , yn , yf , θ
" " yf y1 , . . . , yn !
p(yf | y1 , . . . , yn ) = = =
Z
Z
Z
! *! y
p(yf , θ | y1 , . . . , yn ) dθ
p(θ | y1, . . . , yn )p(yf | θ, y1, . . . , yn ) dθ p(θ | y1, . . . , yn )p(yf | θ) dθ
!*!
" " ! ' (θ, yf ) R " R p(y ) p(θ, yf ) dθ = p(θ)p(yf | θ) dθ
f p(θ) p(θ | y1 , . . . , yn ) ! ' y1 , . . . , yn " ! θ
" * ' ! ! " $ ! * θ ! ! " ! θ * ! ! " ! f
*"
!
%
Y
% !
!
Y ∼ (λ) ! " % p(λ) = 4λ2 e−2λ % % % y Z pYf (y) ≡ P[Yf = y] = pYf | Λ (y | λ)pΛ (λ) dλ Z y −λ 3 2 λ e λ2 e−2λ dλ = y! Γ(3) Z 3 2 !* = λy+2 e−3λ dλ y!Γ(3) Z 3y+3 23 Γ(y + 3) λy+2 e−3λ dλ = y!Γ(3)3y+3 Γ(y + 3) 3 y 2 y+2 1 = , y 3 3 # " % !
3 8 2 = Pr[Yf = 0] = 3 27 3 2 1 8 Pr[Yf = 1] = 3 = 3 3 27
% % & ! % y1 = 3
%
% "
y1 = 3
36 p(λ | y1 = 3) = λ5 e−λ/3 . 5! "
pYf | Y1 (y | y1 = 3) =
Z
% !
pYf | Λ (y | λ)pΛ | Y1 (λ | y1 = 3) dλ 6 y 3 y+5 1 = y 4 4
!*
*"
!
!
6 3 Pr[Yf = 0 | y1 = 3] = 4 6 3 1 Pr[Yf = 1 | y1 = 3] = 6 4 4
% %
!
! %
!
6243 42 −62λ λ e p(λ | y1, . . . , y60 ) = 42! "
!*
%
6 y y + 42 1 62 !*" Pr[Yf = y | y1, . . . , y60 ] = 63 63 y
% % ! % " " % n=1 λ ! " % % ! " % ! n = 60 % " ! ! % λ " % " " %
!
0 •
!
0
•
& "
!
!
*
0.5
0.0
0.1
0.2
0.3
0.4
n=0 n=1 n=60 plug−in
0
2
4
6
8
10
y
" ! y f "
n = 0, 1, 60
*
•
•
0
•
! "
0
&
•
! 0 "
•
!
0
' '
" "
%
!
0
•
0
•
0
•
•
!
" " !
• •
!
!
• •
"
•
' '
! "
!
!
!
" " "
!
!
0 " ! " "
! " " $ !
*
" " ! "
" ! & & " " " ! !
! ' 0 0! ( 0
0 ! ' !
* ! !
!
w = w(y1 , . . . , yn ) &
!
!
w
0
!
w
" " w 0!
0!
y1 , . . . , yn
" ! ' " " ! " !
" ! Xi ! X ,...,X ∼ 1 ( n
(θ1 ) ! % !! ! Y1 , . . . , Yn ∼ ! ! ! (θ2 ) ! θ1 θ2 ! w = θˆ1 − θˆ2 ! E[w] = 0 ! 0 $ ˆ ˆ θ1 θ2 w! ' ! " $ & w # $ " ! $ 0!
*
" ' ! X , . . . , X ∼ ! ! ! (µ , σ1 ) ! " 1 n 1 Y , . . . , Yn ∼
1 ! ! ! (µ2 , σ2 ) ! µ1 = µ2 ! w = µ ˆ −µ ˆ2 ! ' 0 1 ( µ1 , µ2 , σ1 , σ2 w ! # w
! " $ & $ $ " ! 0!
" ! ! " " ! " & 0! !
0! " 0! $ '
! ' & " $ ! ! # " ! ( " ! # ! YC,i $ i YT,i $
i ! ' YC,1 , . . . YC,n ∼ ! ! ! fC ; E[YC,i ] = µC ; Var(YC,i ) = σC2 E[YT,i] = µT ; Var(YT,i) = σT2 YT,1 , . . . YT,n ∼ ! ! ! fT ; "
µC µT σC σT ! '
:µ 0 T
= µC : µT 6= µC
*
$
& w = Y¯T − Y¯C ! ' n 2
+ 2 2 ! ' w ∼ (0, σw2 ) σ = (σ + σ )/n 0 w T C
" 2 " σ 0! ' " 2w 2 ! σT2 ! σ w C σ w $ $ $ " ! 0 0! #
! ' "
! ' ! " ( " ! # ! Y Y $ $ T,i C,i
! ' i i
Xi = '
(
1 0
YT,i > YC,i
X1 , . . . , Xn ∼ ! ! !
p! '
(p)
: p = .5 0
: p 6= .5
P w= Xi ! (n, .5) ! ' (n, .5) 0 w ∼ 0 ! ! w + + n = 100 ! w w 0! + +
! ' |w − 50| !
' $ & ! ! ! ! 0 & w ! !
* +
0.00
0.04
0.08
30
40
50
60
70
w
! (100, .5)
w '
µ0 ! ' " σ0 w
0! ! 0 * " $ " $ ! ( ! " √! ! µ0 = n/2 σ0 = n/2 n = 100 0! ! " w (µ0 , σ0 ) µ σ0 0 0 ! t ≡ |w − µ0 |/σ0 0 $ " 0!
' !
! " # $ % & # $ ' # ' $ = 0.5 # ( # ) * # +
* *
0.00
0.04
0.08
30
40
50
60
70
w
! (100, .5)
(50, 5)
' # # ) $ # $ # * * x1 , . . . , x10 # ( # * * ( = y ,...,y = * # " # " # 1# #10 # xi yi # *
# x1 , . . . , x10 ∼ f
# y1 , . . . , y10 ∼ f
* # $ + µ ≡ E[xi ] µ ≡ E[yi ] # ' # 0
: µ = µ : µ 6= µ
$ * " # ) * # $ ' ' w = |¯ x − y¯| w 0
*
* * $
!
σ
x¯ ∼ µ , √ n σ y¯ ∼ µ , √ n $ # 0
w∼
0,
r
2 2 σ
+ σ n
!
,
$ # # # w SD # $ # '# ' # $ t/σ ≈ 3.2 t # ' t # # ' # # $ ' # 0 0
* # # * *
#
&
( % & &
( % & & &
% ! " # " & $ " & ! % & % ' % & % ' & ( " ' % " ' ' % ' & ' ' ' ' ' ) *+ ' ) & *
## '# '
t
! 0
(
w
*
0.00
0.15
−6
−4
−2
0
2
4
! # " " ! t
6
t! '
* $ ' * # ' * * #$ ' * ) & &%( ' # ' # ' # # ' * # ' # $ ' # ' $ ' ' # # ' # * * ' * # " $ # # * # ' # ' * # ## # ' & )
& & % & & (
!"! ! ! ##$ % &% $ '
*
& ) # ' * '# * $
*
& # ' * * * $
& # # * # '
* ## * * #
% & & * * * ' * ## # * $
( # * # # * * $
!
* * # w % ( & &
' # # n = 147 w = 87 * * # * # # w 0 ! " $ $ $ !! " !! " ! ( #
'
# $ ' # * $ * ' ## $ * $ $ * # * # # # $ 0 # $ # # 0# $
* * # # W * # * $ # * $ # $ * $ # " ' # % % "
!! " & !! " ! ( # !! " & !! " ! ( # " *
& # ' " ! * " # & " # & % % "
" (
# &
"
&
$ $ * '# # # w=9 0 * # * $ * # ! " * $ " ( ) ! $ ! ( # ' " % % " ! ! " & !! " ! ( # $ ! ( # ! ! " & !! " ! ( # $ ! ( # & # ' "
! * " # & " # & % % " " ( # & & (
* $ " * # * '# ' # '# ' # # # * w '# ' # * w 0 $ # $ * ' #
$ # $ * $ # 0 ' ' 0 ' $ " ' $ 0 ' # # ' # ' $ * 0 0 * # # " # $ # ' * * * # # ' # ) # * # # $ * # # # # 0 # * * * 0 # # $ ' $ * #
* # # * *
$ ! ( # " ( ! ( & ! ! " ! ( # $ ! ( # " & $ $ ! ( # "
# ! ) ' ! * $ ! ( # " $ !! % !! " ! ( # $ ! ( # " ! " " ( !! " & $ ! ! % !! " & $ !! % " $ " $ ! ( # " ' " " " ' ' ' $ ! ( # " ' $ & ) ! " ' " # ! " ! " ' ' # ) *+ ' ) & * # ! "
* # # * *
! " & # ' $ ! ( # " " ' $ ! ( # " ' " ! * $ ! ( # " $ !! % !! " ! ( # $ ! ( # "
VIV
0
0
200
200
400
400
OMO
6
7
8
9
10 11
8
10
12
w
w
NYA
WEA
14
16
0
0
100
100
200
200
300
300
5
0
1
2
3
4
5
15
20
w
25
30
w
0
50
100 150
LIN
8
10
14
18
w
0
0
100
200
50
60
70
80
w.tot
! " " (
!! " & $ !! % !! " & $ !! %
" ' $ " $ ! ( # " ' "
! " ! " ( ! " " ! # # " ' ' " ( " " ! ' ! ' ' $ & ) ! " ! ' " ! # ! " ! " ! ' ' # ) *+ ' ) & *
# ! " !
g(x)
!! !
& ( # !
$
`(θ)
p
p ≈ .1
(
" # ! ( & ( # ! " * " # ( & ( # " * * ! !
X
p
p
x1 , . . . , xn ∼ (λ) `(λ) P
x1 , . . . , xn
`(λ)
y , . . . , yn
xi
xi
1
(λ)
ˆ = y¯ λ
" # " ) ( &%( % " & " % & " % * Yij
i j i = 1875, . . . , 1894 j = 1, . . . , 14
Yij
λ
(λ)
ˆ λ
X1 , . . . , Xn ∼
(µ, 1)
d d f (x1 , . . . , xn |µ) dµ dx
d f (x1 , . . . , xn |µ) dµ
d f (x1 , . . . , xn |µ) dx
θ
µ ˆ
=0
=0 =0
y1 , . . . , yn ∼ (µ, 1) `(µ) y1 , . . . , yn P
n = 10
yi
(µ, 1) µ
y 1, . . . , y10 ∼
y , . . . , y10 ∼ 1 `(σ)
Yi
µ µ
(µ, 1)
µ
σ
σ
`(λ)
µ ˆ = y¯
Yi ∼ (λ) λ ≈ 3.1
σ y1 , . . . , y10
y1 , . . .P , yn (µ, σ) 2 −1 2 σ ˆ =n (yi − µ) % " ) " & & " ! i
(µ, σ) y1 , . . . , y10
µ
(µ, σ) `(µ)
y1 , . . . , yn
yi
`(µ)
P[
] = .8
|
|
n
].
T ∼ exp(λ) λ t1 , . . . , tn
λ n = 10 ˆ `(λ) λ λ P
] = .2.
T
Pr[
P[
`(λ) ti
ti
P[H] = 1/4
P[H] = 2/3
P[
]
] = .01
P[
]
]
P[
]
P[
P[
P[
(700, 50) (600, 50)
]
P[
]
P[
]
!
!
"
"
] ≈ .5
0
P[
!
!
!
!
! ' * ' "( +
w
t
w
b
t
y∼
θ
(θ, 1)
y ∼ (m, σ)
θ
y
(θ, σy )
y
θ
y1 , . . . , yn θ
θ∼
(0, 1)
n (m, σ)
θ∼
θ
θ ∼
y ,...,y
1 n
θ
(θ, σy )
% ! "&
=2
0, 1, . . . , 9
(−3.28, ∞)
Pr[yf = k | y1, . . . yn ]
k = 1, 2, 3, 4
(−∞, −.72) (−3.65, −0.36)
% ! "&
=1
X = 122
"( ##
=
"( ##
=
X = 122
nA = 100 nB = 400 nC = 100
.1
.1
.2
!
!
λw = 100
λw
λw
M
M
xi
* ' * "
! &" # &"
x1 , . . . , x15 xi ∼ (λw )
&# & " " !
!
*
! * * &# & " " !
* ' * "
! &"
! '
# &" ! *
*
# *
# *
* *
! *
! *
& (
Y =
X
X
! (
Y
Y
Y
Y
Y
) "
X
X
Y
(
X =
X
X
Y =
X
X=
Y
# ! ) '
%
& (
%
! ( ! ( & # ' ! ( ! & ' & # & ( & ' ! ! & ' ' # )
# ! & ( % " ' & ( ) ) & ' " ) & ' ) ) & & ! ' ' # )
# !
%
) " " # ) ) " ) " ' # ) * ' & $ ' & ! % & ' ( " "" ! ' )
% ( # ! ( ' # ) ' %
X =
X
X
Y
Y
Y
Y
X
X=
X
& ( ! ( ( ) "
Y =
Y
(b) 150 50 0
100
300
60 70 80 90
Distance
temperature
(c)
(d)
0
70 50
60
waiting
80
1
90
0
Manual Transmission
100
ozone
0.6 0.4 0.2 0.0
Acceleration
0.8
(a)
2
3
4
5
1.5 2.5 3.5 4.5
Weight
eruptions
X
Y
$ # $ ' * $ # #
$ # # * # # # * # $ # # * # '# # # # * # # $ # $ # * # # # #$ $ # # * # * X = Y = * # X Y
* # # * *
# ! % ! & ' % ! ' ! & ' ( &
$ * # $ + X Y # '
# ) # ) ' # " # Y X # $ $ *
X ' * * # # # Y
E[Y | X] = g(X) * g ' $ + # g ' * # $ # *
* * g # #$ $ * $ ' * # * # ' $ # # # * # * # $
200 0
100
Draft number
300
0
100
200
300
Day of year
0
100
200
Draft number
300
0
100
200
300
Day of year
! & ""
" ( # " (
* # # * * % ! & % !
# ! ' ' ! & ' ( & & " ! & "" ' & " " ( # " ( ' ' ! & ""
$ * # # "( # " ( * " ! & "" ' * $ "( # " ( ' ' * $ * # * # '
•
# # #$ * # * ! * * ' # ) ' # * # # ' ## # × ×
' * # # ' ' # # * # # ) # ) # # ) # * * # # ' ! & "" ' # * # # # ) $ #
* # # $
# ! ! & ' ! ( % % & ' ! & " &&% $ " & " ! & "" ! &
i yi
(xi , yi )
yi = g(xi ) + i .
i = 1, . . . , n xi
0
5
10
15
20
total new seedlings
25
0
10
20
30
40
50
60
quadrat index
yi
i g
g
yi − gˆ(xi ) r i i
gˆ
ri
y
!
y y x
y
y
x
x x
x
y
x
g
x
y
x
y x
y
y
ri ≡
x
!
g
x
x
i
g(xi )
i
ˆi = ri =
y
y
x
gˆ
g
g
# $ # $ # $ # ' ' $ * # * # * # #
* # # *
" # ) ! % ! $ " ! & " ! % ! $ " # & ' # ) * ' ) ! & "
# $ # * # $ $ $ # # ' # ) # * $ # # # ' # * # ' # #
# (µB , σ) B1 , . . . , B20 ∼
# (µM , σ) M1 , . . . , M17 ∼
# (µP , σ), P1 , . . . , P17 ∼ * " " # " # Bi Mi P # $ ' $ i
µB ≈ 150;
µM ≈ 160;
µP ≈ 120;
σ ≈ 30.
Beef
Meat
Poultry
100
120
140
160
180
calories
'
# B1 , . . . , B20 ∼ (µ, σ)
# M1 , . . . , M17 ∼ (µ + δM , σ)
# P1 , . . . , P17 ∼ (µ + δP , σ) # # $ ' # $ # $ # # # * ' * * # # * * $ # * * * #
* # # * *
# ! ) '* # ! % & " ! % ! $ " ! % ! $ " && ' ' ) ' ' ' ' ) ! & " ' && # ! % & " ! % ! $ " ! % ! $ " & ' ' ) ' ' ' ' ) ! & " ' & # ! % & " ! % ! $ " ! % ! $ " ! ( ' ' ) ' ' ' ' ) ! & " ' ! (
•
•
! % !$ "
# ! % ! $ " ' ' # # $ ! % ! $ " ! & " # ! % ! $ " # & * '
% & " ' # $ + ! #* # ' #* # ' * $ $ # ) #* # * $ #
Beef
50
100
150
200
250
200
250
200
250
calories
Meat
50
100
150 calories
Poultry
50
100
150 calories
µ
µM
µ + δM
µP
σ
!
δP
σ
µP − µB
δM
µM − µB
µ + δP
µB
& $ $ ! ( # * ) % * ( *(
) )
ctrl
trt1
trt2
3.5
4.0
4.5
5.0
5.5
6.0
weight
!
! $ ! ( # ' # ) * '
& $
! & $
" # )
µ δ2
δ1
σ
WC,1 , . . . , WC,10 ∼ (µ, σ) WT1 ,1 , . . . , WT1 ,10 ∼ (µ + δ1 , σ) WT2 ,1 , . . . , WT2 ,10 ∼ (µ + δ2 , σ).
' # # # # # #
Y Y
$ # # * # Y #
# ) # # ) + Y # * # # $ # Y # * # # $ # Y ' # X # # # * Yi Xi " # * * i $ ' Y # #
# &
&& &&
& " ! *( +
*( *
& &
! ( ! (
*% * (
* *
! % ( % % % (
%
$ # ' * # $ # ) ' +
$ # # #$ ! # * (Y1 , . . . , Y54 ) i
* '
X1,i
#
$
X2,i ( 1 X1,i = 0
" # i *
#
(
" # $ i 1 X2,i = * 0 # # ' * # ' X1,i X2,i " # * * # * * * $ i X X $ 1,i
2,i
# ' k k−1 * ' # *
i = 1, . . . , 54
*
Yi = µ + δM X1,i + δP X2,i + i
# (0, σ). 1 , . . . , 54 ∼ $ * $ ' # !
Y = (Y1 , . . . , Y54 )t , B = (µ, δM , δP )t , E = (1 , . . . , 54 )t , $ ' ' ' # 1 1 1 1 X= 1 1
0 0 0 0 0 0 1 0 1 0 0 1 1 0 1
# 54 × 3 # # $ # *
X
Y = XB + E
!
Yi =
X1,i =
(
1 0
X2,i =
(
1 0
i
i
i
,
Y = (Y1 , . . . , Y30 )t B = (µ, δ1 , δ2 )t E = (1 , . . . , 30 )t
1 1 1 1 X= 1 1 1
0 0 0 0 0 0 1 0 1 0 0 1 0 1
Yi = µ + δ1 X1,i + δ2 X2,i + i
Y = XB + E.
XB
µ + δM X1,i
!
Yi + δP X2,i
µ + δ1 X1,i + δ2 X2,i
!
E i
i
+
=
µ + δM X1,i + δP X2,i
µ + δ1 X1,i + δ2 X2,i
(µ, δM , δP )
(µ, δ1 , δ2 )
!
!
X X
X
Y
* $
* # ' +* # $ $ $ #$
* # # # ' ' ! + # ' ### # #* # #
" ) ( &%( & " ) & &
* % & *
( +
# )&
) ! & (
& # %*
$ & # +
&
#
%
(
+ +
(*
+
+(
'
# + #$
#
$ $
#
#
' # $ #
* #$ $ $ $
'
= β0 + β1 + * # # # $ * # * (β0 , β1 )
Y = ( 1 , . . . , 30 )t B = (β0 , β1 )t E = (1 , . . . , 30 )t #
#
1 1 1 2 X = 1 30
Y = XB + E. # # # $
0.25
0.30
0.35
0.40
consumption
0.45
0.50
0.55
30
40
50
60
70
temperature
◦
p
n
Yi = β0 + β1 X1,i + · · · + βp Xp,i + i
Y = XB + E.
Yi ∼ (µi , σ) P
i = 1, . . . , n µi = β0 + j βj Xj,i p+2 (β0 , . . . , βp , σ) `(β0 , . . . , βp , σ) =
n Y i=1
=
n Y i=1
√
i
„
y −(β +
P
β X
)
«2
0 i j j,i −1 1 σ √ = e 2 2πσ i=1 P − n 2 1 P = 2πσ 2 2 e− 2σ2 i (yi −(β0 + βj Xj,i ))
(0, σ)
1 yi −µi 2 1 e− 2 ( σ ) 2πσ
p(yi | β0 , . . . , βp , σ)
n Y
p+2
log
X 1 X yi − (β0 + βj Xi,j ) log `(β0 , . . . , βp , σ) = C − n log σ − 2 2σ i j
C
X 1 X ˆj Xi,j ) Xi,p = 0 ˆ0 + β y − ( β i σ ˆ2 i j X X 2 n 1 − + 3 yi − (βˆ0 + βˆj Xi,j ) = 0 σ ˆ σ ˆ i j
yˆi
(βˆ0 , . . . , βˆp , σ ˆ)
p+1
σ2 p+1 p+1 β ˆ = (Xt X)−1 Xt Y, B
!2
X 1 X ˆ0 + ˆj Xi,j ) = 0 y − ( β β i σ ˆ2 i j X X 1 ˆj Xi,j ) Xi,1 = 0 ˆ0 + β y − ( β i σ ˆ2 i j
i ∈ {1, . . . , n}
yˆi = βˆ0 + x1i βˆ1 + · · · + xpi βˆp .
ri = yi − yˆi = yi − βˆ0 + x1i βˆ1 + · · · + xpi βˆp
i
σ ˆ
X 2 1 X n + 3 yi − (β0 + βj Xi,j ) σ σ i j X 1 n ri2 =− + 3 σ σ i
0=−
σ ˆ2 =
σ ˆ=
1X 2 ri n P
ri2 n
βi
12
β
# # + # # * # # # $ $ # # * # # # ! % ! $ " ! % ! $ " ! & "
•
! % !$ " # &
# #
# # $ " # + • #
# # # # # # # $
• •
$ #
$ # X
•
# # * # ! % ! $ " * # ' # $ *
•
' % * % & # ! % ! $ "
! % ! $ " ! & "
! % !$ " # &
* # ' * ! % ! $ "
& " # & ' % ! % ! $ " !
$ * $ * *
! % ! $ " " " ( " # * * "( ! % ! $ "
! ( ! % ! $ " ! & "
& " %( "
*
* + *( %
!
&%
(
! % !$ " # &
+ %
& ) & "
" & % ! " ( & & ) & # * + ( % + * ! % !$ " # & & * ( + % ! % ! $ " # & ! ( ( ( % *
& *+ & & & ( * *
% & + & & &
$ ) ! % & "
& & & *
& & *
&
*
*
& " %( " % % & ! % + ! * % & $ && " ! &&% ! ( # & ! ( &% ( + + ' % ( " &% " ! ( &% + +
" " ) *+
! % * '
# " ( & ( + & +
# ! & ) & " * & ) & # ! % ! $ " # & & # ! % ! $ " # & ! ( # " & " #
Yi = β0 + β1 X1,i + β2 X2,i + i * # # ' $ # # X1 X2
Yi = β0 + i Yi = β0 + β1 + i Yi = β0 + β2 + i
# # $ #
& ) & # # β0 = = % $ " # & & # ) * # # β1 = ! ! = % $ " # & ! ( # ) * β2 = ! ! = # $ #
! & ) & "
βˆ0 = 156.850 βˆ1 = 1.856 βˆ2 = −38.085
# % ! " ˆ ˆ β β # 0 ! 1 $ ˆ β2
$
βˆ0 ∼ (β0 , σβ0 ) βˆ1 ∼ (β1 , σβ1 ) βˆ2 ∼ (β2 , σβ ) 2
" " ! $ #
$ β0 * #
* # β #
* 1# + β 2 ' # # ' ' * # ' # $ # ' # $ * $
* # # * *
) * + ( ' * ( + ' ( ( " ) % + ' '
# ! ) ' " & ! * & " * ' * # & " * ' & $ % # ! ' % ! ' * '" * ' # & ' & # & "" ! ( ' & !! % ' " & ! & " ' # & " ' & $ % # ! ' % ! ' '" ' # & ' & # & "" ! % & '
& !! % '
" & ! & " ' # & " ' & $ % # ! ' % ! ' '" ' # & ' & # & "" ! % & '
& !! % '
$ ' # & " %( σ
likelihood
likelihood
140 150 160 170
−20
µ
0 10 δM
likelihood
−60
−40
−20
δP
(µ, δM , δP )
" % % & ! # $ σ ˆ ≈ 23.46 # $ ' $ # * *
Y
Y
# ) " + $ * # ' $ % ) " # # # & # ) " # * # $ # + # # # # $ + # $ $ # # $ # '
* # # $
# " ) " ') * '
$ # ' ' ' # ' #
= β0 + β1
*
+
% # ! $ #& # '' ' #& # $ !
σ ˆ ≡# ' &# $ ' # '' ' $ % !$ $ P 1/2 ( r' i2 /(n − p − 1)) $ # & $ & ' $ # n p ! n#$% $ '
400
3.0
4.5
16
22 25
100
400
10
mpg
250
100
disp
4.5
50
hp
4
3.0
drat
22
2
wt
16
qsec 10
25
# "
50
250
) "
2
4
& # ) "
# # # ' * # # ˆ # β0 ≈ 37.3 #
$ ' $
βˆ1 ≈ −5.34 # * # $ # * $ $ * # $ * # * $ $ # ) * # # * # ) ' # $ ' $ * * # = γ0 + γ1 + " # + γ " # γˆ0 ≈ 30.1 γˆ1 ≈ −0.069 * # # # # # * # ) # * $ * * * # * $ # # * # # # # ' # ' ' X # * ' # # # $ # * ' # # # # # $ * ' * # # $ ' ' # ' # ' * # # # * # ' # # ' # # $ $ *
# $ # * $ # $ * # # ' # $ + * # $ + # # $ # $ ' # # $ * * # $ # # $ "( # # $ * σ ˆ σ ˆ ≈ 3.046 * # $ * * # * σˆ ≈ 3.863 * " * * # $ *
* $ # $ * # $ $ * # * # # # $
#$
# # ' % ) "
* #
•
= δ0 + δ1
*
1
+ δ2
2
#
*
+
# # $ * " # # σ ˆ ≈ 2.6 δˆ0 ≈ 37.2 δˆ1 ≈ −3.88 δˆ2 ≈ −0.03 '# # # # $ # #
# $ * $ #
* # # *
# ! ) " ' ) " # $ ' & $ ' # $ # $ * # $ ' % ) " & ) ! & # $ *
#
* # # *
# & ) # ! &% # $ * ' & " % # $ * ' ) ' &% " ( & " ! * ' & " % # & % # ! &% # $ ' & " % # $ '
&% " ( & " ! ' & " % ' %
10
20
mpg
10
20
mpg
30
(b)
30
(a)
3
4
5
50
250
horsepower
(c)
(d) 6 −6
−4
−2
2
resid
resid
150
weight
0 2 4 6
2
10
15
20
25
30
10
fitted values from fit1
15
20
25
fitted values from fit2
−4
0
resid
2
4
6
(e)
10
15
20
25
30
fitted values from fit3
#$
# #
) "
#$
#
#$
δ1
β1
γ1 (−.05, 0)
δ1
δ1
δ2
β
1
β1 ≈ δ1
δ2
# ! ) ' " & ! ( ' * ' & + # ! ' % ! ' % % ' * ' # & ' & # & "" ! & * ' ' " & ! * ' ' & + # ! ' % ! ' +( ' * * ' # & ' & # & "" ! $ * ' ' " & ! ( ' * ' & + # ! ' % ! ' ( ( ' + ' # & ' & # & "" ! % & * ' ' " & ! * ' ' & + # ! ' % ! ' * ' ' # & ' & # & "" ! % & ' '
(−.1, −.04) γ1 6≈ δ2
X Y
γ1
β1
β1
= β0 + β1 + = γ0 + γ1 + = δ0 + δ1 1 + δ2 +
−8
−6
−4
−2
−0.10
−6
−4
−2
−0.10
δ1
0.00
γ1
β1
−8
−0.04
−0.04
0.00
δ2
β1 γ1 δ1
δ2
) "
Y |X
E[Y | X] X Y |X
E[Y | X] Y θ X
Y
#
) * # 2
* ## ' $ $ $ ) '# $ # # 2 * $ # * # * * " # $ # * * $ $ $ # ' $ # # # # X Y # * ' # Y X $ '
$ $ * # $ # # # ' * * # " ! $ "## * # " $ * # $
# * # # ' $ *
* # $ + $ $ * # + # # * * # * # ' # * # " ) & ) & " ) " $ ! " " ( & "" ! " * % ! ) " # '
#
# ' ' $ # # # #
# ' * * # # ' # * # + # # * $ ' # $ # $ # $ # * $ + * # # $ * $ # # * ' $ # * # # # $ ' # Y = X = # # # $ ' # # + * # # $ # # # $ # 37◦
E[Y | X] = P[Y = 1 | X]
E[Y | X] = P[Y = 1 | X]
X X
E[Y | X] = P[Y = 1 | X] =
eβ0 +β1 x 1 + eβ0 +β1 x
0.8 0.4 0.0
pine cones present
(a)
5
10
15
20
25
DBH
0.8 0.4 0.0
damage present
(b)
55
60
65
70
75
80
temperature
β0
β0
β
β1
β1
# ! # ! ! " # ! " # " ! " ! # " # $ % &" " &! " & ' &! " ! ( " # # ) * ) # # ) * # ! ! " # ! " # # # " " # $ ! # " # # " $ & & & ! ( " # # ) * ) # # ) *
+
,- ./ 0. 12 12// . -3 4
i
0.0
0.4
0.8
pine cones present
(a)
5
10
15
20
25
DBH
0.0
0.4
0.8
damage present
(b)
55
60
65
70
75
80
temperature
xi
,- .0
φi
θi = E[Yi | xi ] θi φi ≡ log . 1 − θi
θi
θi =
eφi . 1 + eφi
φi = β0 + β1 xi .
23 21,. 2
φ
,.3 21 - 2,
E(Y | x)
β0 + β1 x θ→1 x → −∞ θ → 0
,
,.3 21 12. 0 -1
x → +∞
β0
β1
4
E(Y | x) β1 > 0
β1 < 0
Y |x
p(Y1 , . . . , Yn | x1 , . . . , xn , β0 , β1 ) = =
Y i
Y i
=
p(Yi | xi , β0 , β1 ) θiyi (1 − θi )1−yi
Y
θi
i:yi =1
=
Y
(1 − θi )
i:yi =0
Y eβ0 +β1 xi 1 1 + eβ0 +β1 xi i:y =0 1 + eβ0 +β1 xi i:y =1 Y i
i
4
(β0 , β1 )
β0
β1
−6
−4
0.35
0.40
0.45
0.50
0.15
0.20
0.25
0.30
β1
−11 −10
−9
−8
−7
−5
β0
4
n
! " # # # # # " ! ' # " & " &" ' # ' # ' # ! (' # ! (' # # # ) * ! " # # ) # #
# ! * ! ! # # " " ! # # "" !
• •
• •
# &&&
&&&
& & &
(β , β ) 0 1
# ! "
#
*
β0
β1
β0 ≈ −9
β0
β1
β1
β0
β1 β0 ≈ −6
+
β1
β
(β0 , β1 )
1
β0
β1
! " & # ! " & ! "
! " # # # #
! " ! " &&& ! " (
" & ! # % &! ' '$ !
&% ' # #! ! & ! & # * * * # &#
&$ % ! # &# # # # % * * * ! " # & '
&&&
" &&& • !
& & &
# &%
& '
&! !
$ # & # &! & ' # ! &
' & ' # &!
# # #
"# $
### # # # ! # '
!! !! # .3 / # # #
•
# #
# #!
#
0 2
!! ! ! "#
• •
& & &
& & &
!
"
y
#
(βˆ0 , βˆ1 ) ≈ (−7.5, 0.36)
•
±2
β1
Y Y
Y
X
Y ∼
(λ)
X
log λ = β0 + β1 x +
λ
! " ! ! #$%&' '()'(*+ !", -,. !" / 0/ 1 ,
/ 2" -,3 , -,4
0 ! 1 !5 /, " / 6!"7 8 !" ! 5 , 2 7
6/7 ! " ! ! " " !, 9 / ! 0 / " ! ! !" ! : 5 " ! ! # /;
! ,+ ! ! !0 8 !" <(λ) 0!, Y 0 ij !0 8 !" 0 ! 1 i ! / j , = > ! 7 ! ! λ 0/ 1 ! ! ! !;,
& & & #
•
•
•
• •
! & , 0 " /
! 0 !/ ! 7 ! ! 0/ ; !" , 9
7 !" !
7 7 ! , ! ! ! # 8 !"+7 1 !0 ! / , !/ 0 5
!, ! & ! " ! ! , 7 ! !" ! ! 0 7 ! 0 , ; 0 !; !
0 !!!" 4 /!" 4 1 4 !" 1 , 9 ! 1 !0 0 7 ! !0 , ! " ! ! , " ! / 6! ;
!", / 0 ! !
, / > ! / ! ! 7 ! , / > ! , ;
/ ! 7 ! 0
4 1 ,
! > ! 7 > ! > ! 2" -, , < ! !
# , / 0 !"7 ! ! ! ! ! , < ! ! , ! ! !
/ !
! # , ; ! ! !" 2" -,3 , , 2" -,
!! /, < ! " ! / , : ! 0 ! / , 7 -7 ! " / . ! 3 , / , ; ! 0/ )
< ! ! "! , ! ! 2" -, !" / 6 ! 0 : ! >, 2" -,
!" ! , # # # # # # # # #
(b)
6 4 0
2
residuals
10 5 0
actual values
15
8
(a)
0.8
1.0
1.2
0.6
1.0
fitted values
fitted values
(c)
(d)
1.2
4 0
2
residuals
6
15 10 5 0
actual values
0.8
8
0.6
0.865 0.875 0.885 0.895
0.865 0.875 0.885 0.895 fitted values
(e)
(f) 3 1 −3
−1
residuals
10 5 0
actual values
15
fitted values
0
1
2
3
4
fitted values
5
0
1
2
3
fitted values
4
5
# #
!" ! 2" -,
! , #
x
β
00 2 , 2/
x
12. 0 2 , 2/
) -, ! ! ", = !5 ! ! 0/ !" 6 0 !
/ : ! ! !/ ! , 2 6! " ! !
! 1 ! -,
7 -, 7 ! -, 7 ! >
, +! /0 7 - . yˆi = βˆ0 + βˆ1 i , - / . yˆi = γˆ0 + γˆ1 i , - . yˆi = δˆ0 + δˆ1 i + δˆ2 i , 9 > " ! !/ ! 5 0 : ! , 2" -, * / , 2" -, * ! " # $ % & $' # (
*
/
(b)
10 0
5
actual values
1 0 −1 −3
residuals
2
3
15
4
(a)
1
2
3
4
year
5
0 2 4 6 8 fitted values
0 −1 −3
−2
residuals
1
2
3
(c)
0 2 4 6 8 fitted values - $
! & > 0 !
/ ! " !" /
7 & > 5 : ! 7 , ! ! & ! & & , + ! -, 6 > " & σ ˆ ! & , &
2" -, *
!" ! , & & & & &
•
> , ! 0 !/ / > 0/ 7 7 !! > ,
- -
y = β0 + β1 x +
,
-
.
x y
4 xf
yf
xf
yf ∼
yf
(µf , σ)
µf = β0 + β1 xf β0 β1
µf
µ ˆf = βˆ0 + βˆ1 xf
(β0 , β1 )
20
10
20
30
20
25
30
10
20
25
10
15
fitted from wt
20
25
30
10
15
fitted from hp
25
30
10
15
fitted from both
10
15
20
actual mpg
10
20
30
- /
10
20
30
µf
ˆ ˆ (β0 , β1 ) (β0 , β1 )
µ ˆf
µ ˆf ∼
σ
xf
(µf , σ )
x µ ˆf y f f µf
µf
y f = µf +
∼ (0, σ)
±2σ
yf
µf
. σ yf , . µ (β0 , β1 ) f
, σ
(β0 , β1 )
,
. - . . . , , , .
,
. , & . , X Y
- - - . , -
0 w w 0
- ,
- -
.
. ,
. , . , . , . ,
+
0
- - - - δP
/
- - +
,
. - -
,
. + -
.
,
-
,
-
. + -
. - , . , . - , . ,
+ - + -
◦ + - / -
β0
β1
- /
γˆ0 ≈ 30 γˆ1 ≈ −.07
σˆ ≈ 3.9
( & & & !( ' ' # !!
- ,
.
µC µT 1 µT 2
. ,
. ,
σ
-
µT 1 = µC
µT 1 = µT 2
.3 2// -0. 23 0 - . Pi = 1 0 , B = 1 0 , i .
b
p
39
40
41
42
43
44
45
HQ
-
,
i
= α0 + α1 Pi + i
.
α0 α1 β1
i
β2
= β1 Pi + β2 Bi + i
. ,
. ,
β1 β2 α0 α1 HQ = 43 HQ = 44 2 σ ˆ =1 ,
. α0 α1 β1 β2
'' '
% ' '
21 23 0 122 , 3 23 - 1 2
,
.
. ,
.
,
. ,
+
4 ˆ ˆ (β0 , β1 ) - - β0 β1 -
/
yi z i z = β0 + β1 yi β1 i ' & $' ( &
,
.
' #
,
.
) ' # +
.
,
SD(βˆ0 ) = .13 SD(βˆ1 ) = .22
βˆ0 βˆ1
# # # ( # # ' # ) ' # & ' # # )
&! ! )
+
#
4
60◦ t1 , t2 , . . . , t100 .
. t1 = , t100 = , 4
y1 , . . . , y100 z1 , . . . , z100
. , t1 , t2 , . . . , t100 yi zi
,
. % !&
yi = β0 + β1 ti + i ,
. % !&
yi = β0 + β1 ti + β2 t2i + i
zi = β0 + β1 ti + i
- -.
. % !&
,
,
,
-
.
/
. % !& , - .
zi = β0 + β1 ti + β2 t2i + i
. % !&
,
,
-
yi = β0 + β1 zi + i
,
.
. % !& , - .
yi = β0 + β1 zi + β2 zi2 + i
,
. , . ,
βˆ0 βˆ2
. , . ,
- . , / ! ! ,
. ˆ ˆ
β0 β1
σ ˆ
. ! !
,
.
, .
, , . .
, .
,
* ) ) #
,
.
.
,
. , . , . ,
& # # # & ( & # * ) ) # # )
yi = β0 + β1 xi + i
β1
yi xi
w = xi + δi δi ∼ (0, .1) i
yi = β0∗ + β1∗ wi + ∗i (β0∗ , β1∗ ) (β0 , β1 ) wi xi
& # # # &# # ( # & # ( # # ( ( & ( & ) # & &# ) * & ) &
&# $ '% # &! ! ' $ & # %
β0
β1 ,
.
x
,
. β0 β1 β . 2 21/
β1
β0
.
(β0 , β1 ) θi = eβ0 +β1 xi /(1 + ,
eβ0 +β1 xi ) θi xi
,
. x β
θ
x β
◦ 36
/ - .
- , . $ ) ,
. , $ ) . ,
X FX / d
fX (x) =
FX (b)
db b=x d 0 F (b) f (x) ≡ F (x) = X X db X b=x A
P[X ∈ A] =
Z
fX (x) dx
Z
f ∗ (x) dx
A
∗ f fX R
R
∗ A
A
f =
A
fX
P[X ∈ A] =
A
∗ fX f
X X -
(# ('
f
P[X ∈ A] =
Z
A
A
f (x) dx
X
fX
-
" # X#
Z
−1 d g (t) pZ (t) = pX (g (t)) dt
pX ! g Z = g(X) ! $
−1
%
A P[Z ∈ g(A)] = P[X ∈
!
A] =
R
A
pX (x) dx
dx P[Z ∈ g(A)] = pX (g (z)) dz dz g(A) R '()* !+ dz '()* !+ P[Z ∈ g(A)] = g(A) & & , −1 pZ (z) = pX (g (z))|dx/dz| Z
-
z = g(x)
−1
pZ (z)
.
X1 Xn n n
~ = (X1 , . . . , Xn ) X
~
X
X1
Xn
~ X
pX~ (x1 , . . . , xn ). n A R
Z
~ ∈ A] = P[X
··· A
Z
pX~ (x1 , · · · , xn ) dx1 . . . dxn
~ X1 ∼ (1) X2 ∼ (1/2) X1 ⊥ X2 X = (X1 , X2 ) P[|X1 − X2 | ≤ 1] pX~ pX~
|X1 − X2 | ≤ 1
A
X 1 ⊥ X2
1 pX~ (x1 , x2 ) = pX1 (x1 )pX2 (x2 ) = e−x1 × e−x2 /2 2
A
P[|X1 − X2 | ≤ 1] =
ZZ
X 1 X2
pX~ (x1 , x2 ) dx1 dx2
A
1 = 2
Z
1 0
Z
x1 +1
−x1 −x2 /2
e
e
0
1 dx2 dx1 + 2
(X1 , . . . , Xn )
Z
∞ 1
Z
x1 +1
e−x1 e−x2 /2 dx2 dx1
x1 −1
.
≈ 0.47 ,
pX~ (x1 , . . . , xn ) = pX1 (x1 ) × · · · × pXn (xn ) (x , . . . , x )
1 n (X , . . . , X )
1 n (X , X ) i j
~ = (X1 , X2 , X3 ) X
P[(X1 , X2 , X3 ) = (0, 0, 0)] = P[(X1 , X2 , X3 ) = (1, 0, 1)] = P[(X1 , X2 , X3 ) = (0, 1, 1)] = P[(X1 , X2 , X3 ) = (1, 1, 0)] = 1/4
X 1 ⊥ X2 X1 ⊥ X3 X ⊥
2
,
X3
X3
.
X1 X2
~ X ~ ≡ (E[X1 ], . . . , E[Xn ]) E[X]
~ ≡ (X1 , . . . , Xn ) X
ij Cov(Xi , Xj )
σii
σ12 σ12 · · · σ1n σ12 σ 2 · · · σ2n 2 ~ Cov(X) ≡ ΣX~ = 2 σ1n σ2n · · · σn
σij = Cov(Xi , Xj )
σi2 = Var(Xi )
σi2
-
~ = E[g(X)]
~ X Z
···
R Z
g
g(x1 , . . . , xn )pX~ (x1 , . . . , xn ) dx1 · · · dxn
~ g(X)
g
X1
X2
Y = X 1 + X2 ! $
! E[Y ] = E[X1 ] + E[X2 ]
! Var(Y ) = Var(X1 ) + Var(X2 ) + 2 Cov(X1 , X2 ) %
!
' %
~ = Y = ~at X
!
! %
P
~a = (a1 , . . . , an ) #
n
ai Xi ! $ P P E[Y ] = E[ ai Xi ] = ai E[Xi ] P Pn−1 Pn at ΣX~ ~a Var(Y ) = a2i Var(Xi ) + 2 i=1 j=i+1 ai aj Cov(Xi , Xj ) = ~ /
!
-
,
.
,
.
k≤n
i = 1, . . . , k
Yi = ai1 X1 + · · · ain Xn =
X j
aij (Y1 , . . . , Yk )
~ Y~ = AX
A
k ×n
aij
~ aij Xj = ~ati X
~ai = (ai1 , . . . , ain )
Yi
~ at X) ~ Cov(Yi , Yj ) = Cov(~ati X,~ j n X n X = Cov(aik Xk , aj` Xj ) =
=
k=1 `=1 n X
aik ajk σk2 +
k=1 ~ati ΣX~ ~aj
n−1 X n X
(aik aj` + ajk ai` )σk`
k=1 `=k+1
Y~ =
-
/
" ~
~ X n E[X] = ~ Cov(X) = Σ A k×n k µ ~ ~ Y = AX ! $ # ! E[Y~ ] = Aµ ! Cov(Y~ ) = AΣA0
'%
- . ,
~ = (X1 , . . . , Xn ) X n fX~ n ~ ~ ~ Y = (Y1 , . . . , Yn ) = (g1 (X), . . . , gn (X)) ~ 7→ Y~ gi g:X fY~ Y~
J ∂Y1 ∂Y1 · · · ∂X1 ∂Xn ∂Y2 · · · ∂Y2 ∂X 1 ∂X n J= ∂Yn ∂Yn · · · ∂Xn ∂X1
|J|
'%
%
J
fY~ (~y ) = fX~ (g −1(y))|J|−1
R R ~ ∈ A] = ··· p ~ (~x) dx1 · · · dxn A P[Y~ ∈ g(A)] = P[X A X
!
~y = g(~x)
P[Y~ ∈ g(A)] =
Z
···
Z
g(A)
pX~ (g −1 (~y )) |J|−1 dy1 · · · dyn
R R '()*+ dy · · · dy '()* + P[Y~ ∈ g(A)] = ··· g(A) & 1 n & , −1 −1 pY~ (~y ) = pX~ (g (~y ))|J| pY~ (~y)
X1 ⊥ X2
X1 ∼ (1) X2 ∼ (2) ~ X = (X1 , X2 ) P[|X1 − X2 | ≤
~ = 1] X |X1 − X2 | ≤ 1 (X1 , X2 )
Y = X − X Y2 1 1 2 Y~ = (Y1 , Y2 ) |Y1| ≤ 1 Y 1 = X1 − X2
Y2 n n R R Y 2 ~ X Y~ Y 2 = X2 ∂Y1 ∂Y1 1 −1 ∂X2 1 = J = ∂X ∂Y2 ∂Y2 0 1 ∂X1 ∂X2
|J| = 1 X1 = Y 1 + Y 2
X2 = Y 2
pY~ (y1 , y2) = e−(y1 +y2 ) × 21 e−y2 /2 =
1 −y1 −3y2 /2 e e
pX~ (x1 , x2 ) = e−x1 × 12 e−x2 /2
2
P[|X1 − X2 | ≤ 1] = P[|Y1 | ≤ 1] =
ZZ
pY~ (y1 , y2 ) dy1 dy2
A
Z Z 1 1 −y1 ∞ −3y2 /2 e e e dy2 dy1 + e dy2 dy1 2 0 −1 −y1 0 Z Z 1 1 −y1 −3y2 /2 ∞ 1 0 −y1 −3y2 /2 ∞ dy1 + e −e e −e dy1 = −y1 0 3 −1 3 0 Z Z 1 0 y1 /2 1 1 −y1 = e dy1 + e dy1 3 −1 3 0 0 1 2 1 = ey1 /2 −1 − e−y1 0 3 3 1 2 −1/2 1−e + 1 − e−1 = 3 3 - . ≈ 0.47 ,
1 = 2
Z
0
−y1
Z
∞
−3y2 /2
/
X2
Y2
X1
Y1
(X1 , X2 ) (Y1 , Y2 )
~ ~ X Y |X1 − X2 | ≤ 1
! "# ! " # $ $ $ $ % "# % " #
&
/
Y Y Y p Y
Y
FY
(P c Y y=−∞ P[Y = y] FY (c) ≡ P[Y ≤ c] = R c Y . p(y) dy −∞
.
,
Y b∈R P(Y ≤ b) = F (b) =
Z
b
p(y) dy
−∞
p(y) = F 0 (y)
Y P[Y = y] = P[Y ≤ y] − P[Y < y] = − FY (y) − FY (y ) , FY (y − ) F (z) . Y z y lim↑0 FY (y − )
( FY (y) − FY (y − ) Y pY (y) = FY0 (y) Y
.
,
FY (y)
Y Y F Y . , . ,
/
Bin (10, .7)
0.0
0.4
cdf
0.20 0.10 0.00
4
8
0
4
8
y
y
Exp(1)
Exp(1)
0.0
0.0
0.4
cdf
0.8
0.8
0
0.4
pmf
0.8
Bin (10, .7)
0
1
2
3
4
5
0
1
2
y
3 y
4
5
/-
&
•
Y MY
MY
(P ty Y tY y e pY (y) MY (t) = E[e ] = R ty e pY (y) Y
' %
,
.
pY
. t=0 ,
M (t)
. Y , t MY (t)
δ>0 MY (t) t ∈ (−δ, δ)
Y
n
MY
E[Y ] =
#
(n) MY (0)
dn ≡ n MY (t) dt 0
/
%
n
!
n=1
Z d d ety pY (y) dy MY (t) = 0 0 dt dt Z d ty = e pY (y) dy dt 0 Z = yety pY (y) dy 0 Z = ypY (y) dy = E[Y ]
d dt
Z
f (t, y) dy =
Z
d f (t, y) dy, dt
f
MY (t)
' % X Y MY ! X (t) = MY (t)
# MX
# M t FX = FY ! ! X Y
! ' % Y1 , . . . MY1 , . . . ! M(t) = limn→∞ MYn (t) !
#
t M(t) #
F
!
!M
y
/
F (y) = lim FYn (y)
F
n→∞
F!
. ,
√
CY (t) = E[eitY ]
i = −1
,
/ ' % #
# X a, b bt
Y = aX + b ! $
%
!
MY (t) = e MX (at) !
MY (t) = E e(aX+b)t = ebt E eatX = ebt MX (at)
' %
X +Y ! $
%
!
X Y Z= ! MZ (t) = MX (t)MY (t)
MZ (t) = E e(X+Y )t = E[eXt eY t ] = E[eXt ]E[eY t ] = MX (t)MY (t)
Y1 , . . . , Yn ! ! ! MY ! X = Y1 + · · · + Yn ! $
' %' !! %
MX (t) = [MY (t)]n
/
,
.
~
pX x1 x2
pX~ (x1 , x2 ) (x1 , x2 )
x1 x2 pX~
,
.
.
,
A
. , N X -
P[X ≥ 1]
(X1 , X2 ) p(X1 ,X2 )
(X, Y ) p (x, y) ∝ ky k > 0 (X,Y ) (x, y) (0, 0) (−1, 1)
(1, 1)
,
.
k
. P[Y ≤ 1/2] ,
. P[X ≤ 0] , . P[|X − Y | ≤ 1/2] , ,
.
~ = (X1 , X2 , X3 ) X
X 1 , X2 , X3
/
. ~ = (X1 , . . . , Xn ) X , X 1 ⊥ X2 X1 ⊥ X3
X2 ⊥ X3
X1 X2
X3
/
X Y (1, 0) (0, 1) (−1, 0) ,
.
(0, −1)
p(x, y) X Y
,
.
U =X +Y
V =X −Y
X = UV
.
. ,
. , . , . , . , . ,
U V p(u, v) U V p(u) p(v) p(u | v) p(v | u) E[U] E[U | V = .5] E[U | V = −.5]
(U, V ) ,
p(x) p(y) p(x | y) p(y | x) E[X] E[X | Y = .5] E[X | Y = −.5]
Y = U/V
X Y
(X, Y )
P[Y > 1]
P[X > 1]
P[Y > 1/2]
X Y
(X, Y )
//
. P[X > 1/2] , . P[XY > 1] , . P[XY > 1/2] ,
t=0
-
Y
MY (0)
n=2
X ⊥Y
-
N
θ θ (0, 1) (N, θ) θ θ
{
(N, θ) : θ ∈ (0, 1)}
- . - . - - . , - . , , ,
( ' ( ! (&# % ( # (' ! " # $ # # # % "" % %& ! $ ! % ! " % $ & # %" % $ " # %
% ' " & " % . ( % "" & % , ! % " . & ! " %" $ % % , % # % , $ " . "" % # % , & % & . ' ! , & " % $ %
/
' # $&
% %"
n
' # % " $
- ' %" %
θ
! !( # &
% # %" !
! "" ! ! # # % (
θ
'
# % ! & (% " # $& % $ $ # "" X %" ' # ! # ( # & " ! % $ # %& % X % ' # " ("$ % # % (n, θ) X∼ (n, θ) X ' ' ' ' $ % ' # & " & " & % ( %" & n ! ' $ "" $ ! # %" % % %& ! %! % n θ θ " % $ '
( $ "θ "% ("$ % ( ! # "% ! & "" ("$ θ ( ! # & X % "" ' $ ("$ # ( ! $ ( " X θ & $ " & % ' % $ "% % $ "% ("$ X =x # %( ! $ # "" " ! ! % ' # " ("$ '
θ
' %
X ∼ (n, θ)
p(x | θ)
n x pX (x) = θ (1 − θ)n−x x x = 0, 1, . . . , n !
%" & " %& % %% ! $ # % "" $ . ! "$% . $ # ' , # ,! % # 1000110 · · · 100 " # $ $ S = {0, 1}n x ∈ {0, 1, . . . , n} # $ $ # ! ' Sx S x 1 n − x 0 # ' % $ "% "" # ( # & s ∈ Sx Pr(s) = θx (1 − θ)n−x s Sx % " ' # % %
%
# #
! $
n
pX (x) = P(X = x) = P(Sx ) =( Sx ) · θx (1 − θ)n−x n x θ (1 − θ)n−x = x
# " & % $ # # ( & ' # n=1 # ! # ( ! % $ # %& % '
% % ' % # % ' X∼ (θ) X∼ (θ) pX (x) = θx (1 −θ)1−x x ∈ {0, 1} %& # # ( " $ & % "" ! % $ "" %" ' $ ! ' X1 ∼ (n1 , θ) X2 ∼ (n2 , θ) X 1 ⊥ X2 ' # # ! % $ $ # X3% = X1 + X2 X3 $ . # % % %" . # %" X3 ∼ (n1 + n2 , θ) n1 + n2 "" # ( # & % " ,$ - . # %" % ,! ! θ # # # % # % % $,& . ! . # " X ⊥ X2 X , $ & % $ ' # % & 1'- # %& " % , # 3 % ' $ % ! # & & % $ ' n=1
X
' %
%
!
Y ∼
θ
X ∼ (n, θ) ! $ n MX (t) = θet + (1 − θ)
%
'
(θ)
#
MY (t) = E[etY ] = θet + (1 − θ).
Pn # %
" X = i=1 Yi "% ' '
#
%
Yi
' '! '
%
(θ)
! "
' % "
X1 ∼ (n1 , θ) X2 ∼ (n1 , θ)
X3 ∼ (n1 + n2 , θ) !
X3 = X1 + X2 ! $
%
#
% "
X1 ⊥ X2 !
!
%
%
MX3 (t) = MX1 (t)MX2 (t) n n = θet + (1 − θ) 1 θet + (1 − θ) 2 n1 +n2 = θet + (1 − θ)
$ "
%
# " # % $ " ""
'
# ! # % & ' ' & # !% $ ' (n1 + n2 , θ) # % & ' ' # % &
#
& " !%
# & # $ " $ " ! $ ' ' # % & ' % # % $ " ! ( # (% ! ! %! ! (
'
' %
X ∼ (n, θ) ! $
! E[X] = nθ ! ! Var(X) = nθ(1 − θ) ! p
! SD(X) = nθ(1 − θ) ! %
# % % ( %" % ' # E[X] X ∼ (n, θ) X = ! Pn % & $$ "" ! ! ' # % # % % ! # X ∼ (θ) Xi X %i=1 i # % & i ' ' $
Var(X) = n Var(Xi )
Var(Xi ) = E(Xi2 ) − E(Xi )2 = θ − θ2 = θ(1 − θ). '
Var(X) = nθ(1 − θ)
"" & & ! " '
# % $ " %
SD(X) & # $ " $ % % # & " !% $ ' %& & & $ # "" # % % & "% $ % % & % & ' # # % ! % $ $ % # % # " # % '
& & & &
&
"! #
&
-
"! #
&
&
&
&
& & & &
& & & &
& &
& & &
&
& &
&
&
$ ! ! $ $
& $
$ % ' # # & " & % ( %" ("$ ! ' x n p # % ! "% % # & " % " " %& " ! '
p
n
# # %" & # % & ' ' '! ' % ' # Y1 , . . . , Yn ∼ P (p) ! % $ # # & # ! % $ ! # %" " %&Y"i & # % & ""X $ # P " % ' n→∞ % ! # & Y" i & % %& " # " # # n p = .5 ' ! # $ ( % $ ! % # %" & # % & p = .05 % # # ! % $ # & % & & % '
$ % ' % ! $ !
Yi
# ( ( ' ( ! (&#% ( # (' # % # !( # $ & % %" %& % "" & & $ # $ %"
n=5, p=.5
0.05
0.0
1
2
3
4
5
0
1
2
3
4
x
n=20, p=.05
n=20, p=.5
p(x)
0.2 0.0
0.1
5
0.05 0.10 0.15
x
0.3
0
p(x)
0.15
p(x)
0.4 0.2
p(x)
0.6
0.25
0.8
n=5, p=.05
1
2
3
4
5
6
8
10
12
x
n=80, p=.05
n=80, p=.5
p(x)
0.02
0.05
14
0.06
x
0.15
0
p(x)
0
2
4
6
8
30
35
x
$ % '
40 x
#
& " &
45
50
$ " % ! $ & % $ # # ( ! ' # # r " $ & % "$ % # % ! & (% " ! ! # ( # N " ! % $ ( & # %& % % '
(r, θ) N∼ (r, θ) % & $# % # # " $ & % %" # # r ! !
, ( & " ! % $ '. & " & " % N#+ # ' # % ! $ & % $ " # ! " " % $ " ' # ' # $ & % "$ % $ " # % ! & r=1 ! #N( & % ! % $ # %& ! # ( % r=1 N % % ' $ ' % ("$ % θ N ∼ (θ) θ N ( ! # & "" & "" ("$ % ( ! # "% ' # θ N θ % " $
pN (k) = P(N = k) $ # % %" = P(r − 1 k+r−1 ! # %" $ k + r ) k+r−1 r θ (1 − θ)k = r−1 %
k = 0, 1, . . .
'
''' ! '''
N1 ∼ (r1 , θ) Nt ∼ (rt , θ) N1 Nt ! ! # # % ' # & $ %" " # P # ( P $ ' # $& % "$ % % (N + r ) ri N # #1 $ & % "$ % % # # $ i i ' ' ' r1 N1 + · · · + Nt P # $ & % "$ % # $ ' ( ! # r1 + · · · + rt P N≡ Ni % # # $ $ % ! # % % # '
r≡
' %
r(1 − θ)/θ ! 2
%
!
ri
Y ∼
N∼
(r, θ)
$ % ( # % $ " %
E[Y ] = r(1 − θ)/θ
'
r=1
(r, θ)
# # % $ " %
Var(Y ) =
r>1
""
""
# % % $ & !
E[N] = =
∞ X
n=0 ∞ X n=1
# % & ' ! ' ' %
r=1
n P[N = n] n(1 − θ)n θ
= θ(1 − θ)
∞ X
n(1 − θ)n−1
n=1 ∞ X
= −θ(1 − θ)
n=1
d (1 − θ)n dθ ∞
d X = −θ(1 − θ) (1 − θ)n dθ n=1
d 1−θ dθ θ −1 = −θ(1 − θ) 2 θ 1−θ = θ = −θ(1 − θ)
#
% %
# %& ! %(( # # #
%! %
/
$& & ! ! %(( "" $ $ " ' 2
E(N ) =
∞ X
,
% '
n2 P[N = n]
n=0
= θ(1 − θ) = θ(1 − θ) =
∞ X
n=1 ∞ X n=1
(n(n − 1) + n) (1 − θ)n−1 n(1 − θ)n−1 + θ(1 − θ)2 ∞ X d2
1−θ + θ(1 − θ)2 (1 − θ)n 2 θ θ n=1
∞ X n=1
n(n − 1)(1 − θ)n−2
∞ 2 X 1−θ 2d = (1 − θ)n + θ(1 − θ) 2 θ θ n=1
1−θ d2 1 − θ + θ(1 − θ)2 2 θ θ θ 1−θ θ(1 − θ)2 = +2 θ θ3 2 − 3θ − 2θ2 = θ2 =
# % %
Var(N) = E[N 2 ] − (E[N])2 =
1−θ . θ2
# $ % % # # ( & " ! % $ % ! ' $ % ' ! " # ( & " ! ! ""$% # $ ' $ % ' % ! $ ! # #
"" '
" # "#
0.020
0.00
0.04
0.08
r = 1, θ = 0.1
0 20 40
probability
probability
probability
N
r = 5, θ = 0.1
60 100
350
0.010
20 N
r = 30, θ = 0.1
200 N
' % $
0.000
0.05
0.02 0.06 0.10 0.14
0.0
4 6
0.05
0.15
r = 1, θ = 0.5
2
14 probability
0.02 0.06 0.10
0.4
N
8
probability
0.2
0
4
45
r = 5, θ = 0.5
0 N
r = 30, θ = 0.5
30
N
probability
0.25
0.2
0.4
0.6
0.8
1.0
4
2.0
r = 1, θ = 0.8
0.0 N
2
N
12
r = 5, θ = 0.8
0
6 N
r = 30, θ = 0.8
2
& " & (
0.03
15
#
0.01
0.006
probability
0.002
probability
probability
-
& " # "# & " # "# &
# " ' # $ $ ' $ % # # " # " / !%
! & % # "& # •
-
# & $ " & " ! % $ %" # & " ! % $ # ' "" # & " ! % $ " # # $ & " ("$ # & $" & " !% $ " # # %" # ' & & " % $ & %" # & % # " $ &
" " %" # ! & % ! % & $ $ "" %& " % & % ! %! % " % & ' % # & % ! # % $ "$ % % % ! ' % $ % $ ! # $ % # % $ ! % & & & ( % # % % '
!( ($ ! % ( !&
% & % #
&
# % '
$
% "" # $ ( % "" # % " %
# ($ & #
& ( %" (% ' ( % % # # & % " ! % " ' # % $ ! % ! % " & # # # {a, c} ! ' # % "" ! # % ' a c # % & " ! % ! %" ' & $ # & " # ( # '
' !(# ($ ! $ ( $ ! ! #
" # % % % # % # $ " & % # % % $ ! ! ! '
-
# $ # $& % $ & # ' # % % k " $ & # # % $" ( % # % # $& % y1 , . . . , yk yi ' & # $ & $ % ! ! # $& % %" i y1 + · · · + yk = n ! # % " # % # p ≡ (p1 , . . . , pk ) k n $ & % %" ' % $ " ' % $ "% Y ∼ (n, p) Y ≡ (Y , . . . , Yk ) ( % " # ' $ ( % 1
k
Y
E[Y ] = µ = (µ1 , . . . , µk ) ≡ (E[Y1 ], . . . , E[Yk ]) = (np1 , . . . , npk ). # # %! % ! & (% " % # ' $ i Y Y # $ $ & % & i $ & $ %% ! %" ! % $ i i n % '. '. Yi ∼ (n, pi ). , , "# $ # # % "" & " # % ! ! ' % "" Yi # # & $ ! ! ' # Y1 = n Y = ··· = Y = 0 Yi # % & 2 # # k ! " !% $ ( Y2 , . . . , Yk # # % & % ( ! # % ' Y1 ' %
Y ∼
(n, p)
fY (y1 , . . . , yk ) =
%
n y1 ···yk
n py1 · · · pykk y1 · · · yk 1
& $ " & "
n y1 · · · yk
n! =Q yi !
%& % %% ! $ # % # % # " % ! abkdbg · · · f # $ & ! ( ! $ " %" ' $ # $ ""
!
# # %" & $ " & " n $ $ & $ #
· · a} |b ·{z · · }b · · · k · · k} |a ·{z | ·{z y1 y2
yk
# % " # % $ "% $ ' ' ' # # & % " ' a yk k
fY (y1 , . . . , yk ) = (
$ &
% $ # $
Q
)×
pyi i
Y
'
( %
pyi i
=
$ #
y1
Y n pyi i . y1 · · · yk
-
' %
Y ∼
(n, p)
p∗i = pi /(1 − p1 ) %
% / '
(Y2 , . . . , Yk | Y1 = y1 ) ∼
(n − y1 , (p∗2 , . . . , p∗k ))
i = 2, . . . , k !
!
$ % #
& $" & " !% $ % ! ' ' ( % % " ' ! % & " m # % $ " & % ' # "$ & ! % # "$& $& k × m ' # $ % ! ! %& ! & '
n
#
k
! % $ $ ! & ! " $ #
• • • • •
#
% ! & $ ! $ $ ""
( %
# #
"
"" $ '
% & '
& " % ! & " # ! & '
% $ ! %" %
# #
( % '
% ! (% ( % # ! & '
#
$ % ( " ! `1 $ % ( # % " '
`2
! #
# " $& % ( # % # ! & ' # y Y ! % $ # % %& % % ' # &
Y ∼
λ
e−λ λy y! # & ! %( ! # %
%
pY (y) =
y = 0, 1, . . .
% '
E[Y ] = λ.
Y ∼
%
(λ) ! $
(λ)
t
MY (t) = eλ(e −1)
%
! ∞ X
tY
MY (t) = E[e ] =
ty
e pY (y) =
y=0
=
∞ X y=0
e e−λ (λet )y = −λet y! e
=e
Y ∼
%
(λ) ! $
ety
e−λ λy y!
y=0 ∞ −λ X −λet
λ(et −1)
∞ X
e
y=0
(λet )y y!
Var(Y ) = λ
%
$ % $ "" % ( # # % & ! #! # & & % $ '
2
E[Y ] =
∞ X
y=0 ∞ X
y2
e−λ λy y! ∞
e−λ λy X e−λ λy = y(y − 1) y + y! y! y=0 y=0 = =
∞ X
y=2 ∞ X z=0 2
y(y − 1)
e−λ λz+2 +λ z!
=λ +λ
e−λ λy +λ y!
Var(Y ) = E[Y 2 ] − (E[Y ])2 = λ
'
% ! % "
d2 E[Y ] = 2 MY (t) t=0 dt d2 λ(et −1) = 2e dt t=0 d t λ(et −1) = λe e t=0 hdt i t t = λet eλ(e −1) + λ2 e2t eλ(e −1) 2
t=0
=λ+λ
2
Var(Y ) = E[Y 2 ] − (E[Y ])2 = λ
'
%
YiP ∼ (λi ) iP = 1, . . . , n
%
Yi n n Y = Y λ = λ Y ∼ (λ) ! ! 1 i 1 i ! $ %
# % & ' ! ' # ( !
# #
MY (t) =
Y
# & #
$
MYi (t) =
(λ)
Y
t
t
eλi (e −1) = eλ(e −1)
! % $ '
# $ & % ( $ %% i = 1, . . . , n Yi ! & ' $
# % ! ! # % Di Y i ∼ (λP i) ! ! ' # $ & D%i ( % Yi ' Y = Y D = ∪Di P # " # $ $i # # % ' # Y ∼ (λ) λ= λi % & ' $ % $ # ( % # % %% " # ! ! ! Y # ( # ! % $ ' # % $ ! (λ) Y ∼ (λ) # ! ( !$ " ( # $ % % ! & " ! ( ! ! Y ! %! & " ! % $$ # %& % # Y1 Y θ 2 ! ! ' Y1 ∼ Y ∼ (λ(1 − θ)) Y1 ⊥ Y2 $ %(λθ) ' % # ' # 2 # & % = 1, 4, 16, 64 λ
& " % " %& " ' # λ $ # % & ' !
# % & ' "" $ # %" & # % & ' # Y Pλ # # % #∼ (λ) ' & $ % Y Y = ∼ (1) λ % # % ' i=1 #Yi # %"Yi & # % & "" $ # "" Y
% & " %& " # "% ' λ %
# %" $ # ! % $ $ α $# % %! ! % '
% " , "$ & $ " $ % % ! $% ' α
0.04
pY(y) 0.08
0.0
0.1
0.2
0.3
0 2
λ=1
y 4
λ = 16
10 15 20 25
y
6
pY(y) 0.02
0.04
0.00
0.05
0.10
0.15
0.20
0 2 6
λ=4
4 y
λ = 64
8
50 60 70 80
y
λ = 1, 4, 16, 64
0.00
pY(y)
% & ' % $
0.00
pY(y)
* (* *( '+ * * + ( ' ' ( * ()' ' )' )' ' (* ' &()' '*+' *(' & ' ( ()'* 20 ! " (* + ()' (* ' ' *((' * (* ' ( ' α &
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& &* + &( ()' - + * + ( ' * ' ()' * ( * (* (* ' ' * * α , ' + * + ( '(*&& &* ( ()' & '()' ( ()' * (* & & '( '( .' ' ()' ' * ' ()'* ' ) ' * & & " )' ' * (* * (' *() * &
& & & / )* ) ' * * ' ' ) ) (' ( ' ' ( ' ' * & )* ' ''& )' * (* & (* ' ' & (' * ()' 0 &()' & ' * (* & (* & ' * + ( * &(' 12 * (' ' ' (' & & & " )' & * + ') ' * * ( (* ()' ' ( ( * ' )' ' ' +* ' * ()' ) & * ( & * ' ' & ( ()' * (* (*& ' *'0 * (' & 4 &' & & & && & 3 & & (
* (' 5 6
& * 86 * * () 47 () *
4 3 4 8 7 2 8 3 7
6 47 7 68 4 76 6 6 8 8 47 8 9 7 : ' +' 4 ' ' +'
8 74 8 8 8 9 7 * (' &
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: )' ' ()' ' * ' ()'* ()' '(* ' ( & & " )' * ( * (* (* ' * + ( ()' * *( α &
&
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' &()'
( ( * *( () (
(* ' '& ' ' * ()' & ' *&(' * +*' xn −x n α & & * )' ' ) ' ' )* ) & ) ' & *(* ' 'n! e n (& )' ' * '(' * ' (* +& +' & ' ∞ x * (* ' * (' * ' )' (* * * * &+ ()' & * *( & (* ' * ()' + * ' *(' & ()' ( ' ' α (' n ()' ()' *& ' ' ( (' # $ )* ' * (* ' 4 ) ()'* ( : & & & ()' '*+' ' ) * 5 & ! " ' *' ' ()' ( ' ' ' * ( ()' ' ( )* ) ' + * ' ' (' &* ) &* ( ( & & & & 0 & " ' ) ' '' ( * (' * (' * &
3 8 &
12 )* ) 6 (* ' ' ' (' ()' ' * (' 6
() (& ( α (* ' 3 α 4 & * * + ()' *( * '' * &(' & 8 ( * * + (* ' ()'& & &' (* ' () ( 689 2 (* '&' ' ' ' 3 44 &)' ' (' & & ' ()'α ' (* & ()' * & 4 84 * & & 0 8 * ) * ()' '*+' ' ()'* ( ()' '(* *% * *( *() ! ( )* ) ' ' ' *+ ' 4 )'* & ( * (* )' * &5 & 7 & " ( * ' '' () ( ()' ) ' ()' ' ) ' * ' ( ' * ' ) ' % ' ( & ' ' ' ( ' () ( ()' * ( * (* ()' & * *( (* ' * (* ' * * & + '' ' ( *() α() ( ()' & (* ' & ' '*((' ( :& ( ()'* α ' * + * (* ' & ' ()' ' ' ( ' *(' ' ( (' (* &+ ()' & & & * *( & &' % ') ' '()
& * ' &* ' ('& * * + ()' * (* * (*(*' ( & & & ' * & &&
α
! ! % % # $ ' "
" ! #
! %
α
400 300 200 0
100
Number of Groups
500
0
2
4
6
8
10
12
Number of Particles in Interval
$ %
% %! ! % $ %
" $ " % " ! % $ '
& % # % " "!
$#
α
% "
$ '
# ! % $ %& ! % $ # !% $ # ( $ " # # % ' % 1, . . . , n ' # & Y ∼ (1, n) ' p(y) = 1/n
% ' # ! % $ %& ! % $ $ ! & ! " % y = 1, . . . , n & " ! % "" % # % %& # # # $ & % ! & ! $ "" " " ' # " %& % ' "" $ $ " ! % $ % " % $ $n ! ""$ % & " ' %
" ! & " % '
# $ $ $ %& ! % $ # ! % $ # ! ( % # %(" ' % [a, b] ' "# $ # # & # $ ! # # ! % Y ∼ (a, b) $ %& # "" ! # # & ' # ! p(y) = 1/(b − a)
% ' # & (% ! & & % $ % " y ∈ [a, b] % ' $
%( % ! & & " % & ' # y1 , . . . , yn (a, b) # & '" ' ' ˆ # !
(ˆ a, b)
p(y1 , . . . , yn ) =
(
0
n 1 b−a
a ≤ y(1)
# %
# # & & ! $ (a, b) # $ & # ! ' #$
!
b ≥ y(n)
& ""
b−a ! ˆ ' b = y(n) a ˆ = y(1)
"
# $
% % " % & # & " $ ! !
Γ
Γ(α) =
& & ' # !
R+ Z ∞ 0
tα−1 e−t dt
"
%& $ # & & $ ! % % ' % $ % $ % #
$ ! & # & %
%
%
(
Γ(α + 1) = αΓ(α) α>0 % ( % Γ(n) = (n − 1)! n √ Γ(1/2) = π
$ & %
α
p(y) =
!
# & &
1 y α−1e−y/β Γ(α)β α
% & ' Y ∼ (α, β) $% ' # & & β
β
α, β)
%
•
y≥0
! % $ % ("$
'
•
! % $ #
α
! '
! $ % ("$
# " $ % ' # $ %( % ! % # ( ! % α
# ' & & "" ! # %& % # & & α ! % $ '
# $% " " ! " % # ' ' ' # $ % $ %( # $ ! % # % & # " # ( # & α = .5 " ' # ! % " %% ! ! % ("$ ' % β # % "" ! %& % ' !% " % & ! ! $ 'β # " %& % $ β p(y) y " # % $ # # % ' # ! " %& % & ! ! y/β # % & ' ' ' % & % " %& % '
$ % ' % ! $
!
# ""
'
! " ! "# $ %& " " " " ! % %& ' "&#" ( ) $ " *%+,$ - ,'. $ ," *%+
β=1 1.5
3.0
β = 0.5
1.0
α = 0.5 α=1 α=2 α=4
0.0
0.5
p(y) 0.0
1.0
p(y)
2.0
α = 0.5 α=1 α=2 α=4
2
3
4
5
0
4
6
8
y
β=2
β=4 0.4
y
α = 0.5 α=1 α=2 α=4
0.1 0.0
0.0
0.2
0.4
p(y)
0.3
α = 0.5 α=1 α=2 α=4
0.6
2
0.2
1
0.8
0
p(y)
0
5
10 15 20 y
$ % ' & &
0
10 20 30 40 y
! % (% $ ("$
α
!
β
'
) ' )$ "&#. & %& & %& & # ) "" *%+ . ". ) & .) " %&".. *%+ )" ) "#& )$, $ & * & &+ "#& "#
!!
X∼ # % & ' '
(α, β)
pX (x) =
Y = cX x = y/c
!
Y = cX
Y ∼
(α, cβ)
1 xα−1 e−x/β , Γ(α)β α
dx/dy = 1/c
1 (y/c)α−1e−y/cβ cΓ(α)β α 1 (y)α−1e−y/cβ = Γ(α)(cβ)α
pY (y) =
# #
#
! ' "
&
% '
(α, cβ) # & & ! (% % % %! ! #
!!
Y ∼
(α, β)
( %" # % & '
E[Y ] = αβ
Z
∞
1 y α−1 e−y/β dy α Γ(α)β 0 Z 1 Γ(α + 1)β ∞ y α e−y/β dy = α+1 Γ(α) Γ(α + 1)β 0 = αβ.
E[Y ] =
#
y
" $ " "" $ ! # % ! Γ(α+1) = αΓ(α) & & ! # %" '
#
" % # % % % ! ! ! "$ ! # # %" ( % $ $" ' % '
! !
Y ∼ (α, β) ! MY (t) = (1 − tβ)−α t < 1/β
!!
Z
∞
1 y α−1 e−y/β dy α Γ(α)β 0 β α Z ∞ ( 1−tβ ) 1−tβ 1 α−1 −y β y e dy = β βα Γ(α)( 1−tβ )α 0
MY (t) =
ety
= (1 − tβ)−α
Y ∼
(α, β)
Var(Y ) = αβ 2
SD(Y ) = !!
•
αβ.
% '
$ # •
√
# (
! "
#
$
# " & & # & $ " ! ( #
# & $ ! & " % " ! % $ % $ # $ #
! ! % $ ' # & $ " # & ! % # ( Y $% ' #
" ! % $ # % & # ! Y λ>0
Y
pY (y) = λ−1 e−y/λ
%
y≥0
'
! %
' # ! $% ! $ % ' % Y ∼ (λ) $ % ' % $ % ("$ ' #
" ! % $ # "
λ
# & & ! % $ # ' # & ! & % α=1 SD ( # % & ' ' ' #
" ! # & & $ & ! ! % & y=0 "" ' # ("$ ! %& # ("$ ! # % λ p (0 | λ) ! % ' $ "" $ ' & "" ("$ Y % ( ! % "% λ y ("$ "% ("$ % ( ! % & "" ("$ '
λ
λ
10
y
0
2
4
p(x)
6
8
lambda = 2 lambda = 1 lambda = 0.2 lambda = 0.1
0.0
0.5
1.0
1.5
2.0
x
$ % '
" !
/
%
%
6 6 238 8 2 8 7 % 6 239 α % 64 2358 + .. α %&
# %. %&8 "%
Y
) : & λ( % 0
λ m P[Y ≤ m] = % P[Y ≥ m] = 0.5 m
Z
m
λ−1 e−y/λ dy = 0.5.
0
λ log 2 % % 6m = 4 λ 6 9 7
8 2 % 6
7 73 % 4 λ 8 87
! !
" ! ! ! (λ) # ! # ! $ ! $ " T
$ ! " " " t T ≥t S ! ! " " " # S = T −t S !
T > t
r>0
P[S > r | T ≥ t] = P[T ≥ t + r | T ≥ t] = =
P[T ≥ t + r, T ≥ t] P[T ≥ t]
λ−1 e−(t+r)/λ P[T ≥ t + r] = = e−r/λ . P[T ≥ t] λ−1 e−t/λ
S (λ)
$ ! " #
"
t $ # ! $ ! # ! " #
$$ (λ)
! # "
%
T
+
, ) /
% % $ ! ! " $ $ !
$ ! $ $ "
$ $$ $ λ
" ! # ! # $ ! T
(λT )
" # ! ! # $ I I2 I1 1 # ! # $ "
I2
$ $ # ! ! ! "
! $ ! " ! # t0 T1 $ # ! $ " t0 Y1 = T1 − t0 T1 Y1 ! " # !
y
Y1
Pr[Y > y] = Pr[
$
$ #
[t0 , t0 + y]] = e−λy
! "
Pr[Y1 > y] = e−λy ⇒ Pr[Y ≤ y] = 1−e−λy ⇒ pY (y) = λe−λy ⇒ Y1 ∼
$ #
t0
(1/λ)
! $ $ ! # $ T2
" #
Y2 = T2 − t0
#
Y2
y>0
$
Pr[Y2 > y] = Pr[ [t0 , y]] $ $ = Pr[ [t0 , y]] + Pr[ [t0 , y]] = e−λy + yλe−λy #
!
pY2 (y) = λe−λy − λe−λy + yλ2e−λy =
λ2 ye−λy Γ(2)
" ! $ ! Y (n, 1/λ) " #$ !n # # ! "
Y2 ∼
% %
α ≡
(2, 1/λ)
P
Y1 , . . . , Yn Y ≡
αi
X
Yi ∼
(α, β)
Yi ∼
(αi , β)
!!
$
"
! " ! ! Yi
! " αi
β
% ! ! 2 $ β = ! α = p/2 #p # ! " 2"
%
p
% % X ∼ χ2n !!
!
Y ∼ χp
Y1 , . . . , Yn ∼
(0, 1)
$
X =
" "
P
Yi2
! α β (α, β) # ! " Y
pY (y) =
Γ(α + β) α−1 y (1 − y)β−1 Γ(α)Γ(β)
! " (α, β) Y ! " "
∼
% %
Y ∼
(α, β)
E[Y ] = Var(Y ) = !!
"
$
"
#
(α, β)
"
y ∈ [0, 1]
! $
α α+β αβ (α +
β)2 (α
+ β + 1)
! " $ # ! ! " # ! ! $ # # α β " # ! " α>1 β>1 p(y) (1, 1) ! "
(0, 1)
# $ # ! " " " " " x1 , . . . , xn ∼ (0, 1) $ (0,1) $ # # #
x(1) x(1) # " FX(1) (x) = P[X(1) ≤ x]
= 1 − P[ Xi
= 1 − (1 − x)n
# pX(1) (x) =
x]
Γ(n + 1) d FX(1) (x) = n(1 − x)n−1 = (1 − x)n−1 dx Γ(1)Γ(n)
$ " # $ (1, n) $ " % % % % % $ ! ! # ! "
% %
X1 ∼ Y ≡
!!
(α1 , β) X2 ∼
X1 ∼ X1 + X2
(α2 , β)
X 1 ⊥ X2
(α1 , α2 )
$ "
! " ! X X # 1 2 "
β
• • •
!
"
! !
! ! ! $
!
8
a
4 0
2
p(y)
6
(a,b) = (0.3,1.2) (a,b) = (1,4) (a,b) = (3,12) (a,b) = (10,40)
0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.6
0.8
1.0
y
4
b
2 0
1
p(y)
3
(a,b) = (0.3,0.3) (a,b) = (1,1) (a,b) = (3,3) (a,b) = (10,10)
0.0
0.2
0.4 y
6
(a,b) = (0.3,0.03) (a,b) = (1,0.11) (a,b) = (3,0.33) (a,b) = (10,1.11)
0
2
4
p(y)
8 10
c
0.0
0.2
0.4 y
" ! " ! " ! "
•
"
" $ $ # ! ! " $ ! ! # ! $ ! " ! ! $ #
! ! "
! $
%
µ∈R
σ>0
p(x | µ, σ) = √
!
"
1 x−µ 2 1 e− 2 ( σ ) . 2πσ
% % % % %
4 2
8
) µ ≈ 8.08 σ ≈ 0.94 - 8
- ◦ ◦ 44 46 ◦ 7 3 ◦ −21 −19
0.2 0.0
density
0.4
4
6
8
10
12
temperature
" ! ◦ ◦ " $
19 − 21
! 44 −46◦ " " "
% 8◦ ◦ / 45 20
/
t 2 2 7 8.5◦ 9.0◦ /
2 2 7 ◦ t 8.5 ◦
9.0
P [t ∈ (8.5, 9.0]] =
Z
9.0 8.5
√
"
1 t−8.08 2 1 e− 2 ( 0.94 ) dt ≈ 0.16. 2π 0.94
4 4 ! & ! : &
% & ! $ & $ &
& , • $ * + $ * + $ % $ •
+ + +
$ * $ *
$ *&+ $ *&
19/112 ≈ 0.17
8 2 2 7
, . % 2 2 7
◦ ◦ 7.5 8.0
P [t ∈ (7.5, 8.0]] =
Z
8.0 7.5
√
1 t−8.08 2 1 e− 2 ( 0.94 ) dt ≈ 0.20. 2π 0.94
4
8
2 2 7
% %
Y ∼ (µ, σ)
MY (t) = e
σ 2 t2 +µt 2
.
15/112 ≈ 0.13
!!
Z
1 2 1 e− 2σ2 (y−µ) dy 2πσ Z 1 2 2 2 1 √ e− 2σ2 (y −(2µ−2σ t)y+µ ) dy = 2πσ Z 2 (µ−σ 2 t)2 µ2 1 2 1 − 2 √ = e 2σ e− 2σ2 (y−(µ−σ t)) + 2σ2 dy 2πσ
MY (t) =
=e =e
% %
ety √
−2µσ 2 t+σ 4 t2 2σ 2
σ 2 t2 +µt 2
Y ∼
(µ, σ)
E[Y ] = µ !!
.
Var(Y ) = σ 2 .
!
E[Y ] = MY0 (0) = (tσ 2 + µ)e
σ 2 t2 +µt 2
t=0
$ 2
MY00 (0)
E[Y ] =
2
=σ e
σ 2 t2 +µt 2
2
2
+ (tσ + µ) e
σ 2 t2 +µt 2
= µ.
t=0
Var(Y ) = E[Y 2 ] − E[Y ]2 = σ 2 .
= σ 2 + µ2 .
$ ! ! "
! " ! # $ # ! " ! $ ! $ ! " $ " "
(0, 1)
% %
X ∼
Y ∼
(0, 1)
(µ, σ)
Y = σX + µ
X = (Y − µ)/σ
Y ∼
(µ, σ)
X ∼ (0, 1)
!!
"
X ∼ (0, 1)
Y = σX + µ
"
!
MY (t) = eµt MX (σt) = eµt e "
Y ∼ (µ, σ)
X = (Y − µ)/σ
"
σ 2 t2 2
"
MX (t) = e−µt/σ MY (t/σ) = e−µt/σ e
σ 2 (t/σ)2 +µt/σ 2
t2
=e2
$ " $ 2 $ $ χ # ! ! " 2 $$ ! χ # # ! " ! "
"
% %
X ∼ χ2n !!
Y1 , . . . , Yn ∼
$
n=1
(0, 1)
X =
"
Z
y2 1 2 MX (t) = E[e ] = ety √ e− 2 dy 2π Z 1 1 2 √ e− 2 (1−2t)y dy = 2π Z √ 1 − 2t − 1 (1−2t)y2 −1/2 √ = (1 − 2t) dy e 2 2π = (1 − 2t)−1/2
tY12
!
"
X∼ (1/2, 2) = χ2 # 1 " n>1 MX (t) = MY12 +···+Yn2 (t) = (1 − 2t)−n/2
X∼
!
(n/2, 2) = χ2n
"
P
Yi2
~ X
! " Σ~ X
n
! ! $ !
µX~
$ $
~ ! ! # X
pX~ (~x) =
1
1
(2π)n/2 |Σ|
1 2
t
e− 2 (~x−µX~ ) Σ
−1 (~ x−µX ~)
"
# ! # ! " ~ ∼ (µ, Σ) " |Σ| X ! # " " Σ # # ! " ! $ $ ! $ # $ $ 2 " Σ σ $! ! #! ! ! $ $ $ !
2 σ1 0 0 σ22 Σ= 0" 0" ""
··· · ·" · "" ""
0 0 "
""
· · · σn2
$
pX~ (~x) =
1
1
1 2
t
e− 2 (~x−µX~ ) Σ
−1 (~ x−µX ~)
(2π)n/2 |Σ| n Y 1 Pn (xi −µi )2 n − 1 1 i=1 σ2 i = √ e 2 σ 2π i i=1 “ ”2 n Y x −µ 1 − 12 iσ i i √ , = e 2πσ i i=1
$ # ! ! # $ n ! " # ! Xi " $ " i ∼ (µi , σi ) X ! ! " · · · = σn = 1 Σ n In σ1 = ~ ~ µ1 = · · · = µn = 0 X ∼ (0, In ) X ! ! ! " n
# ! ! X1 X2 X1 ⊥ X2 ! " Cov(X1 , X2 ) = 0 Cov(X1 , X2 ) = 0 X 1 ⊥ X2 # ! ! $ " X1 X2
" " # (X1 , X2 ) ∼ (µ, Σ) Cov(X1 , X2 ) = 0 X ⊥ ! $ !1 "
% % ~ ! ! ! X = (X1 , . . . , Xn ) ∼
(µ, Σ)
X2
" #$
Σ !
Σ11 012 ··· 01m 021 Σ22 ··· 02m Σ = 0m1 · · · 0mm−1 Σmm
Σii ni × ni 0ij ni × nj ! Pm ni = n 1 ! ~ ! ! ! Σ Y~i Y~1 = (X1 , . . . , Xn ) Y~2 = X ~ 1
(Xn1 +1 , . . . , Xn1 +n2 ) Ym = (Xn1 +···+nm−1 +1 , . . . , Xnm ) νi ν1 = (µ1 , . . . , µn1 ) ν2 = (µn1 +1 , . . . , µn1 +n2 ) νm = (µn1 +···+nm−1 +1 , . . . , µnm )
~
Yi
~ Yi ∼
! !
(νi , Σii )
!! # !
~ → (Y~1 , . . . , Y ~m) # ! X
pY~1 ,...,Y~m (~y1 , . . . , ~ym) = pX~ (~y1 , . . . , ~ym) = =
(2π)
1 Q m n/2
i=1
1
1
(2π)n/2 |Σ| 1
|Σii |
1 2
e− 2
=
1 2
t
e− 2 (~y−µ) Σ
−1 (~ y −µ)
Pm
m Y
yi −νi )t Σ−1 yi −νi ) i=1 (~ ii (~
1
1
1
ni /2 |Σ | 2 ii i=1 (2π)
t
e− 2 (~yi −νi ) Σii
−1
(~ yi −νi )
! ! ! $ # ! $ " $ $ " " pX~ {~x : pX~ (~x) = c} c $ # ! t −1 xi (~x − µ) Σ (~x − µ) # ! $ $ $ "
pX~
Σ
P (~x − µ)t Σ−1 (~x − µ) = n1 (xi − µi )2 /σi2 pX~ (~x) = c # $ $$ $ ! µ " σi /σj
! " " ! " # $ ! $ # $ ! ! # ! " $ E[X1 ] = $ " E[X2 ] = 0 σX1 = σX2 = 1 σX1 = " 1; σX2 = 2 σ 1 = 1/2; σX2 = 2 $ ! $ X $ $ # ! " $ ! $ ! ! $ "
"
$ # $ "
)
) ".
$ ' "&#. $ ' "&#. & & $ & & $ & . & & &. $ $ & . $"% )"% %& $" $ %& $ * + )" $ %& $ * + & &
". $"% )"% %& $" $ %& $ * + )" $ %& $ * + $ & &. $ $ & . $"% )"% %& $" $ %& $ * + )" $ %& $ * +
& & & . &
&
". $"% )"% %&
2 −4 0
2
4
−4
0
x1
x1
(c)
(d)
2
4
2
4
2 −4 0
2
4
−4
−2
0
x1
x1
(e)
(f)
2 −4
0
x2
2 0 −4 −4
−2
0 x1
2
4
−4
−2
0 x1
" ! " " E[X1 ] = E[X2 ] = 0 " σX1 = σX2 = 1 " σX1 = 1; σX2 = 2 % " σX1 = 1/2; σX2 = 2 % $ # " ! # ! "
4
4
−2
4
−4
2
0
x2
2 0 −4
x2
−2
4
−2
4
−4
x2
0
x2
0 −4
x2
2
4
(b)
4
(a)
$" $ %& $ * + )" $ %& $ * + & $ . & & &. $ $ & . $"% )"% %& $" $ %& $ * + )" $ %& $ * +
& &
&
". $"% )"% %& $" $ %& $ * + )" $ %& $ * +
•
$ ! # #$ X X # $ $ 1 2 ! ! "
•
& & !
! &
ij
pY~ (~y ) =
Σ
1 (2π)n/2 |ΣY~ |
! # & & " *%+ , & * + "
" 1
1 2
t
e− 2 (~y−µY~ ) ΣY~
−1
(~ y −µY~ )
Y ∼ (µY~ , ΣY~ )
,
~ ∼ (0, I ) " X ~ $ $ # (0, 1) ! X " $ n # $ ! $ "
(n − 1) " 1/2 ~ ~ pZ~ = pY~ Z = Σ X +µ
# ~ Y~ ! # Z ! ! ! $ ! # ! # ! ! ! $ " ! " " $ # # ! # !
pZ~ = pY~ ~ |Σ|1/2 # ! # Σ " # Z pZ~ (~y ) = pX~ Σ−1/2 (~y − µ) |Σ|−1/2 1 −1/2 t −1/2 = √ n e−1/2(Σ (~y−µ)) (Σ (~y−µ)) |Σ|−1/2 2π 1 1 t −1 = e− 2 (~y−µ) Σ (~y−µ) n/2 1/2 (2π) |Σ| = pY~ (~y )
~ X
$ ! ! ! ~ Y ! # ! # ! ! " ! ! ! " " ! " # $ ! $ # $ ! ! # ! " $ E[X ] = $ 1 " E[X2 ] = 0 σ1 = σ2 = 1 σ1,2 = 0 "
σ1,2 = .5
"
σ1,2 = −.8
$ #
& . %
)
$ "
) ".
$ ' "&#.& . $ ' "&#.& . %# ) %# * + %# * + %# * + &
. . %$ & . & .
% %& %# %# *% + %# %& " %# . %$ %& %& & . %& & . $ $ * + $ * +
'. , %, . %# , & . * + $ , . $ , %# %& , $ &. $ $ & . $"% )"% " " $" $ %& $ * +
2 −4 0
2
4
−4
0
x1
x1
c
d
2
4
2
4
2
4
2 −4
0
x2
2 0 −4
0
2
4
−4
−2
0
x1
x1
e
f
2 −4
−4
0
x2
2
4
−2
4
−4
0
x2
−2
4
−2
4
−4
x2
0
x2
0 −4
x2
2
4
b
4
a
−4
−2
0 x1
2
4
−4
−2
0 x1
" ! " " E[X1 ] = E[X2 ] = 0 σ1 = σ2 = 1 " σ1,2 = 0 " σ1,2 = .5 % " σ1,2 = −.8 % $ # " ! # ! "
)" $ %& $ * + %&".. * ,% + . %$ & , % % %# ,
". * + * + $"% )"% $" $ %& $ * + )" $ %& $ * + %&".. * ,%+
$ $ $ ! ! ! ! # "
% % ~
X ∼ (µ, Σ)
A Y ∼ (Aµ, AΣAt ) !!
n
! " "
! n
! !
Y = AX
n
pY~ (~y ) = pX~ (A−1 ~y )|A−1 | 1 t −1 (A−1 ~ − 12 (A−1 y ~ −µX y −µX ~) Σ ~) = 1 e n/2 (2π) |A||Σ| 2 1 t −1 (A−1 (~ y −AµX y −AµX − 12 (A−1 (~ ~ )) Σ ~ )) = 1 e n/2 (2π) |A||Σ| 2 1 t −1 )t Σ−1 A−1 (~ y −AµX y −AµX − 12 (~ ~ ) (A ~) = 1 e n/2 (2π) |A||Σ| 2 1 t t −1 (~ y −AµX y −AµX − 21 (~ ~ ) (AΣA ) ~) = 1 e n/2 t 2 (2π) |AΣA |
$ $
(Aµ, AΣAt )
"
n
(µ, Σ) A n n b
t Y = AX + b Y ∼ (Aµ + b, AΣA )
!!
$
~ ∼ X
"
! ! !
! !
n
¯ 2 X ¯ ⊥ S2 X) !!
X1 , . . . , Xn ∼
(µ, σ)
S2 ≡
! $
Y~ = (Y1 , . . . , Yn )t ¯ Y 1 = X1 − X ¯ Y 2 = X2 − X
Pn
i=1 (Xi
−
"" "
¯ Yn−1 = Xn−1 − X ¯ Yn = X
# # " " "
S2
# $ #
(Y1 , . . . , Yn−1)t ⊥ Yn
" # "
Pn
S 2 ⊥ Yn
(Y1 , . . . , Yn−1)t
" " # $ #
Yn
"
"
"
P ¯ = 0 " # (Xn − X) ¯ = − n−1 (Xi − X) ¯ " (X − X) i=1
i=1 i #
S2 =
n−1 X i=1
¯ 2+ (Xi − X)
# $ #
n−1 X i=1
(Y1 , . . . , Yn−1 )
¯ (Xi − X)
t"
!2
=
n−1 X
Yi2 +
i=1
n−1 X i=1
Yi
!2
"
1 − n1 − n1 − n1 · · · − n1 −1 1 − 1 −1 · ·" · −"n1 ""n "" n ""n "" "" " ~ " " ~ ~ Y = X ≡ AX 1 −n − n1 · · · 1 − n1 − n1 1 1 1 1 ··· n n n n
!
$
A # $ 2 t " ~ Y ∼ (Aµ, σ AA ) n−1 A " #
t
AA =
Σ11 ~0 ~0t 1/n
! ! ~0 (n−1)×(n−1) (n−1) $ # " ! " " (Y , . . . , Y )t ⊥ Y
Σ11
1
n−1
n
! ! # !
"
t
t
"
F
! # $ ! # ! " " " " X1 , . . . , Xn ∼ (µ, σ) µ σ ! " ¯ ! " ! µ µ ˆ=X √ ¯ ∼ (µ, σ/ n) X
t
¯ −µ X √ ∼ (0, 1) σ/ n
$$ $ ! √ ! # ! " " $ ! µ µ ±2σ/ n ! ! # ! $ "
!
σ
−1
σˆ = n
P
¯ 2 (Xi − X)
1/2
σ
¯ −µ X √ ∼ (0, 1), σ ˆ/ n
√ ! " $ $ # ¯ −µ)/(ˆ (X σ / n) ! ! " # ! " ¯ " ! " # 2 X ⊥ σ ˆ S = P ! ! " 2 2"
nˆ σ =
%
¯ (Xi − X)
= W1 + W2 V1 ⊥ V2 V = V 1 + V 2 W ! W1 ⊥ W2 V W V1 W1
! ! V2 W2
!!
! !
# $
MV2 (t) = MV (t)/MV1 (t) = MW (t)/MW1 (t) = MW2 (t)
T
F
% % X1 , . . . , Xn ∼ ¯ 2
(µ, σ)
S2 =
X) !!
S2 ∼ χ2n−1 . 2 σ
V =
2 n X Xi − µ σ
i=1
V ∼
χ2
Pn
i=1 (Xi
−
.
n
n X ¯ + (X ¯ − µ) 2 (Xi − X) V = σ i=1 ¯ 2 n n X X ¯ 2 X −µ Xi − X ¯ − µ) ¯ +n + 2(X (Xi − X) = σ σ i=1 i=1 2 ¯ 2 n X ¯ Xi − X X −µ √ = + σ σ/ n i=1
≡
S2 + V2 σ2
S 2 /σ 2 ⊥ V2 V =
V2 ∼ χ21
σ
W1 ⊥ W2 W1 ∼ χ2n−1
! ! " "
2 n−1 X Xi − µ i=1
"
+
Xn − µ σ "
2
W2 ∼ χ21
≡ W1 + W2
$ $ #
T ≡
n−1 n
¯ √ ¯ n(X − µ)/σ X −µ √ =p . σ ˆ/ n S 2 /(n − 1)σ 2
# ! " T " U/ V /(n − 1) U ∼ (0, 1) V ∼ χ2n−1 U ⊥V $ # # ! " " t n−1 T ∼ tn−1 ! " "
p
"
r
% % U ∼ p 2
U/ V /p ∼ χp
(0, 1) V ∼ χ2p
p
− p+1 )p 2 2 ) Γ( p+1 Γ( p+1 2 2 2 √ = pT (t) = t + p p p √ Γ( 2 ) π Γ( 2 ) pπ !!
U ⊥ V
T ≡
− p+1 2 t2 . 1+ p
U T =p V /p
Y =V #
! # !
(U, V ) → (T, Y ) ! # " # ! T 1
XY 2 U= √ p
V =Y
$
dU dT dV dT
#
(U, V )
dU dY dV dY
21 Y√ = p 0
Y 12 = √ p 1 1
T Y√− 2 2 p
p u2 1 1 −1 − v2 2 e . pU,V (u, v) = √ e− 2 v p Γ( p2 )2 2 2π
# #
(T, Y )
1
2 t2 y p 1 1 −1 − y2 y 2 y pT,Y (t, y) = √ e− p e √ p p Γ( p2 )2 2 2π
(T, Y )
T
F
! #
pT (t) =
Z
T
pT,Y (t, y) dy
1 =√ p√ 2πΓ( 2p )2 2 p
Z
∞
y
p+1 −1 2
y t2
e− 2 ( p +1) dy
0
p+1 2 p+1 2p ) Γ( =√ p+1 √ 2 t2 + p πΓ( 2p )2 2 p Z ∞ y p+1 1 −1 − 2p/(t2 +p) 2 y e × dy p+1 2 0 2p p+1 Γ( 2 ) t2 +p 1
− p+1 )pp/2 2 Γ( p+1 2 2 = t +p p √ Γ( 2 ) π − p+1 2 ) Γ( p+1 t2 2 = p √ 1+ . Γ( 2 ) pπ p
" " # # # ! t " (0, 1) " ! ! ! $ t ! ! ! ! " (0, 1) " ! $ p→∞ tp (0, 1) " $ $ $ " $ "
" "
$ # "
$ ' "&#. & %& . $ . $ ! . $ . $ ! & $ . ". $ & .) " )" & %.) $" . ".) " "#& $ ) ! ".) "#& . ! ! "
0.4
0.2 0.0
0.1
density
0.3
df = 1 df = 4 df = 16 df = 64 Normal
−4
−2
0
2
4
t
"
t
# # # # !
(0, 1)
# $ " " √ ¯ n(X − µ)/ˆ σ ! " ! " # (0, 1) √ $ $" ¯ − µ)/ˆ n − 1(X σ tn−1 " " ! " t ! $ #$ (0, 1) σ ! " " " ! n σ $ " #
T ∼ tp
Z
) Γ( p+1 2 E[T ] = t p √ −∞ Γ( 2 ) pπ ∞
"
− p+1 2 t2 dt 1+ p
! −p $ " t→∞ t # # " p > 1 t ! ! " ! ! " 1 p > 1 E[T ] = 0
! ! $ # # " tp p > 2 " ! ! p > 2 Var(T ) = p/(p − 2) T k " # # k
E[T ] < ∞
"
p>k
F
!
"
" # !
! ! # $ "
" "
# # "
" ! ! # # # $ " ! # ! " # ! $ # ! "
# !
# ! "
! # ! " # ! # ! $ # ! "
$ # ! # $ # ! $
! ! $ ! " # ! # $$! # "
"
! $ # # ! " $ $ $ " " " # # (3) ! "
(n, p)
n = 13
p
! ! ! $ #$ # "
! # $ $$ "
# $ ! $ "
$
# ! "
"
$ #
! #
$
%
! $$ !
$
"
" "
"
$$ "
! ! $ # ! # $$ "
"
! " # # ! (λ) $ " ! ! ! (3, 22) "
$ " ! " $ ! $ $ # " # # " ! # ! $ $ "
$
! # ! ! # ! ! #
"
#
! #
! # ! $$
! # !
! # ! ! # !
# ! # !
$
# ! # $ # # ! " ! ! $ $ " " " " # $ $ " ! $ # # ! " $ $ $ " " " # " (3) (λ) λ ! $ $ ! "
"
! # ! ! # ! $ ! # ! ! # ! $ ! # $ # ! $ $ ! $ $ $ ! $ " $ !
# $ " $ ! $ $ ! ! # $ " ! # ! $ $ " $ $ # !
$ " ! # $ $ ! $ "
$
! # #
$ # ! #
# ! $ $$ "
" "
!
"
! !
! " E[Y 2 ] ! " "
! " E[Y 2 ] # $ " "
# $ " $
! !
" " % % % % $! ! # # ! ! # ! ! " " ! $! $ $ "
$ ! # # ! ! #
$
$ ! # # ! !
!
! # ! # # !
!
! # ! ! # ! ! " # ! " # # ! $ ! ! "
# ! " # # ! $ ! # ! $ "
"
! ! # $ $$ $!
# $ # ! $! " $ $ # $$ $ # ! ! # ! "
! $ ! " $ $ ! ! # ! "
$ # ! # ! $
$
! # ! # ! $ # $$
$
# ! ! ! ! "
"
$ ! ! $ # ! " $ $ ! ! $ # $ # ! # ! $ $ $ " ! # $ ! " # " # # ! $ $ # $ # "
$ ! # $ ! ! $ $ ! ! # # " # # ! # $ $ $ $ $ ! " $ $ $ $ " $ $ $ $ $ $ ! "
$ !
$ $ #
"
$ $ # $ $
"
# $ # $ $
! ! $ # $ # " $ ! $ $ $ # ! $ ! # # $ # ! $ ! " # $ # ! # #$ "
"
#
% % % % % % ! ! ! # ! $ "
!
$ # !
"
# ! $ ! $ " # " " "
"
$ $ #
$ $ "
X ∼ (λ) " Pr[X ≤ 7] = "
" "
X
P7
−λ x λ /x! x=−∞ e P7 −λ x e λ /x! Px=0 7 −λ x λ /x! λ=0 e
# ! "
Y Pr[X ≤ .5|Y ≤ .25] = "
"
.5 .25
" $ # ! # !
$ X ∼ ! (µ, σ 2 ) " Pr[X > µ + σ] " ! "
" "
"
" $ # ! # !
"
X , . . . , X ∼ (0, 1) " X¯ ≡ (X + · · · + X )/100 " Y ≡ (X + 1 100 1 100 1 " $ · · · + X100 ) "
¯ ≤ .2] Pr[−.2 ≤ X Pr[−.2 ≤ Xi ≤ .2] " Pr[−.2 ≤ Y ≤ .2] " ¯ ≤ 2] Pr[−2 ≤ X " Pr[−2 ≤ Xi ≤ 2] " Pr[−2 ≤ Y ≤ 2] " ¯ ≤ 20] Pr[−20 ≤ X " Pr[−20 ≤ Xi ≤ 20] " Pr[−20 ≤ Y ≤ 20] X ∼ (100, θ) " P100 f (x|θ) = θ=0 "
"
" !
" $ # ! # !
# X
"
Y
"
f (x) =
R1
f (x, y) dx R01 f (x, y) dy " R0x f (x, y) dy 0 "
f (x, y)
"
X , . . . , X ∼ ! ! (r, λ) 1 n
! "
f (x1 , . . . , xn ) = "
" "
"
P Q [λr /(r − 1)!]( xi )r−1 e−λ xi Q Q " [λnr /((r − 1)!)n ]( xi )r−1 e−λ xi P Q " [λnr /((r − 1)!)n ]( xi )r−1 e−λ xi
" $ ! $ ! " ! "
!
" "
$
# " " " # ! k=2 ! # # ! " " ! ! # # Y1 Y # " 2 Y1
" " "
%
%%
Y ∼ (1, n) ! n " " ! " " " "
! Y! =y ∈6 {1, 2, . . . , n} nˆ p(y)
y 6∈ {1, 2, . . . , n}
! $ ! # ! #$ ! $ $ # ! " " " ! " " " " ! T ! # ! ! # $ " ! $ # $ $ ! " " " "
Tˆ
"
Y
$ "
$ !
E[Y ]
"
Var(Y ) MY (t)
Y ∼
(a, b)
"
"
"
$ # ! # $ " ! $
(1, ∞)
$ # ! # ! $ "
" "
!
" "
x1 , . . . , xn ∼
" " "
$ "
"
!
" "
"
!
"
"
(0, 1)
$ $ # % "
"
!
"
(−∞, ∞) # ! " "
#
x(n)
" #
"
! ! n $ $ ! $ # ! n " ! $ " " " " ! ! X X !1 n " " " """ "
µ1
µn
σ1
σn
# $ X " i # X " " " X " 1 n $ $
"
" "
# ! #
! $ $ $ ! #
" "
" #
x∈R
X1
"""
Xn
"
x2 1 lim ptp (x) = √ e− 2 p→∞ 2π
# # ! p (x) t p tp # ! " ! $ " $ x % % % % " "
# # # " ! $ ! # ! $ $ # # $ ! $ $ ! ! " $ $ ! $ $ ! ! $ ! ! # # " ! $ # $ $ # " $ ! $ $ # θ $ ! " $ " ! θ θ ! # ! $ $ # ! # ! # ! ! " $ # $ $ $ θijk i " $ j k i j $ $ " # # µ {θijk }k µ ! ij # $ ij" ! k
{θijk }k | µij , σk ∼
" " "
(µij , σk )
# $ $ i $ " # # # $ µi {µij }j µi " $ $ $ j µi # ! ! µ
σi
" ! $ " " "
{θijk }k | µij , σk ∼ (µij , σk ) " " " {µij }j | µi, σj ∼ (µi , σj ) " " " {µi }i | µ, σi ∼ (µ, σi )
$ # ! "
"
"
%
% % %
$ # $ $ ! # #
% % % ! $ ! $ # ! !
! # ! # ! " ! $ " $ $ ! $ ! " $ ! $ $ $ $ ! "
" ! $ ! " # # ! "
$
% % ! # # ! ! ! $ # ! $ $ " ! ! ! ! # # #! # ! # "
! $ $ $ # 2 ! ! ! ! ! $ ! # $ $ "
$ $ # ! $ " ! ! " " ! "
% % ! # ! $ ! ! ! "
%
"
%
! # #
! #
# # ! # "
#
% ! # $ $ " ! # $ "
! ! # " $ ! # ! # ! " $ ..
% .$.
. " "
$! ! $ $ ! " $ " $ $ # # $ ! " # # $ " # ! ! $ # ! ! " $ ! Y1 , . . . , YT # "
Yt
" $ #
"
!
". . . %& $" % )" .
". . %& & )"
". . . . * + %& )" "%&# %
". . " . %& &# % )" &. ") .
". . ")&$ %& & %& )&$ )" .
%&#
". . % &. %& % &. )"
"
". . . ". * % %"" + %& %
20 40 60 80
1960 1970 1980 1990
DAX
UK Lung Disease
1992
1994
1996
1998
1500
Time
monthly deaths
Time
2000
1974
1976
Time
Canadian Lynx
Presidents approval
7000
Time
1820
1860
1900
1945
1955
1975
1965
1975
1980
1985
250
Sun Spots
0
60
200
UK drivers
1970
1980
Time
number of sunspots
Time
deaths
1978
30
0
320
36.4
CO2 (ppm)
Mauna Loa
0
trappings
Closing Price
Temperature
Beaver
1750
1850
Time
1950
Time
" % % ! # $ ! ! $ $ $ #
$ $ # $ $ ! 2 % % ! # ! $ ! ! ! ! #
% % % # $ ! ! "
$ ! ! # ! " •
" # $ ! ! ! ! # # ! ! " " " Y Y $ $ " $ t t+1 " # Yt+1 Yt " $ $ " ! $ !
# #$ ! $ #
! $ " # $ # $ ! $ " # "
! !
" # "" " " $ " $ " $ " " # $ " #
! !
Cor(Y , Y ), Cor(Yt , Yt+1 ), . . . , Cor(Yt , Yt+5 ) # $ # t t " " " " $ $ " " " $ #
"
%& #' %& '
% '
% ( ' %& "' %& '
% '
% ( '
30 50 70
yt+1
37.0
Presidents
36.4
yt+1
Beaver
36.4 37.0
30
yt "
Yt+1
60 yt
Yt
#
" $ #
"
& & " & % & ' & % ( '
% ' % ( '
$
! $ $ ! ! " $ # ! $ ! # ! ! " " $ # $ ! " $ $ ! ! $ !
lag = 1
37.2 36.4 36.8
37.2
36.4
lag = 2
lag = 3
37.2 36.4
36.8
Yt+k
36.8
36.8
37.2
36.4
36.8
37.2
lag = 4
lag = 5
37.0 36.6
Yt+k
37.0
37.4
Yt
37.4
Yt
36.6 36.4
36.8
37.2
36.4
Yt
0, . . . , 5
37.2
Yt
36.4
Yt+k
36.4
"
36.8
Yt
37.2
36.4
Yt+k
36.8
Yt+k
36.8 36.4
Yt+k
37.2
lag = 0
Yt+k
36.8
37.2
Yt
Yt
#
k =
$ $ $ $ $ ! $ $ $$ # # $ $ " ! # $ ! # " ! # "
% %
! $ ! $ # $ " $ # " # $ # {Yt } $ # Yt+1 Yt " " $ Yt−1 Yt−1 Yt+1 Yt $ $ " " " " $ # " " $ #
"
& & % ' % & ' % & '
% & '
% '
% ( '
$ Yt+1 # # Y t " $ ! Yt $ $ # " # Y Yt−1 Y $! t+1 $ t # $ "
! " ! Yt−1 ! Yt+1 Yt−1 ! $
! !
" $
" " #
#
" " & " " & " "
! ! " $ Cor(Y , Yt+k | Yt+1, . . . , Yt+k−1 ) # ! " " t $
Yt 36.4
36.6
36.8
37.0
37.4
36.4 36.8 37.2
36.4
36.8
37.2
Yt+1
36.4
36.8
37.2
36.4 36.8 37.2
37.2
36.4 36.8 37.2
Yt−1 "
$ #
Yt+1
# $ #
Yt−1
Yt
#
! $ " Yt+1 ⊥ Yt−1 | Yt " ! # # ! Y = β0 + β1 Yt + t+1 ! " $ ! $ t+1 " # $
# ! " "
& !
" $ $
" " "
" ! !
" Y = β0 + 0.8258Yt + t+1 ! $ # " " t+1 " ˆ β0 = 36.86 t " $ # √ $ Yt+1 ∼ (36.86 + .8258Yt, .012) ! $ # " $ $ ! $ ! " ! $ ! $!
Yt =
!
+
$ $
+
.
$ ! ! ! # ! ! ! $ $ " ! # ! ! # t # $ "
t
2
gˆ(t) =
.5yt−6 + yt−5 + · · · + yt+5 + .5yt+6 12
"
! ! " t $! ! $ ! " " $ gˆ " # ! " &
g(t)
gˆ
" " $ ! " # # ! ! # ! $ # ! ! 2 # ! " " " " ! $ " gˆ g ˆ + # ! "
" $ #
"
& & & & & & (
$ $ # ! $ $ "
Yt t # $ # ∗ Y = Yt+1 /Yt $ $ t $ ! ∗ " Yt Y " # $ $" " t ! Y ! $ $ "t ! # " ! # $ Yt − Yt−1 ! $ $ $$ " " Yt Yt−1 " Yt Yt−1 # # Yt = β0 + β1 Yt−1 (β0 , β1 ) = (0, 1) (β , β1 ) # " 0 $ $ ! # " Yt ≈ Yt−1 # " !
Yt −Yt−1
(b)
−2
0
resids
340
−4
320
co2
2
360
4
(a)
1980
1960
1980
Time
Time
(c)
(d)
340
co2
0
320
−3 −2 −1
cycle
1
2
360
3
1960
2 4 6 8
12
1960
Index "
gˆ
gˆ
1980 Time
2
!
Yt ∼ (Yt−1 , σ)
! # "
1995
1992
1995
(c)
(d)
4000
−200
6000
−3
−1
" $ $ " ! # # "
Yt
Yt−1
1
3
Theoretical Quantiles
DAXt−1
1998
0
Time
Sample Quantiles
Time
5000 2000
0
1998
2000
DAXt
1992
−200
5000
DAXt − DAXt−1
(b)
2000
DAX
(a)
Yt − Yt−1
" $ #
Yt
Yt − Yt−1
"
% ' & % & '
%& ' %& ' % & '
% ' " %& ' %& '
•
$ ! ! # $ $ # ! " % ' & % ' % ' & % ' $ $ $ $ ! %& ' & %& ' " ! ! ! $ ! # " ! % ' & % ' %#' & % ' ! " $ $ # ! "
" # " ! " Yt∗ ! # $ " ! # ∗ ! # $ Y ∗ − Yt−1 " t ∗ " Yt∗ Yt−1 ∗ ∗ ∗ ! # ∗ Yt Yt−1 Yt ⊥ Yt−1 " # ∗ " ! Y ∗ ! t# # Yt ∼ (µ, σ) " ! # ∗ " Y " ∗ ! ! " Y ∼ (1.0007, 0.01)
" $ #
"
% ' & % & ' %& ' %& '
% & '
% '
! # ! ∗
Yt ≈ Yt−1 ∗ Y ∼ (1.0007, 0.01) Yt
500
0
Yt
1500
(d)
0.98
1.04
0.92
1.00
(c) Sample Quantiles
Time
−3
−1
" ! # ∗ Yt # ∗ ∗"
Yt−1
1
3
Theoretical Quantiles
ratet−1 " ∗
500
Time
1.00 0.92
0.05
1500
0.92
ratet
0
−0.05
1.00
ratet − ratet−1
(b)
0.92
rate
(a)
Yt
∗ Yt∗ − Yt−1
! " ! $ " $ # # ! " $ $ " # $ # " $ ! " $ ! ! # ! # " ! # ! $ " $ $ # $ ! $ ! ! $ $ ! $ ! $ "
%
%
!"
#$ % & '
( ) & *
+,-+./012 3 ,40+5 .3 5,/ 5 1214, 60/575 819+05 15 0 13 +05
!:
;< = $ ' *$*
!>
( ? & @ $** A = & **
B
C D E F B
G H I J I !
?%&
B
DN
M
A $# K L & *
!
< #$ %
L & ? $ % K L & *
205.471+5 204 5 ,95 +,85 . 5 7120 6 6 6 5 , ,0 .0/85 8058./- /05804 3 ,40+5 ,8 025 !
&= $*& *
1 0 +,85 1/1+1-,75 8, .-7205 1/4 ,3 780 178, ,220 +18.,/5 1/4 ./802 208 ,2 8 0 ,8 02 41815085 ./ .-720 9 1 0 +,85 1/1+1-,75 8, .-720 ,3 780 128.1+ 178, ,220+1 8.,/5 1/4 ./802 208 ,2 8 0 ,8 02 4181 5085 ./ .-720
620180 1/4 8 1 -,,4 3 ,40+ ,2 9,4 803 02187205 , 8 0 50 ,/4 901 02 50 8 0 4181508 1 4,05 718.,/ 1 021-0 , 02 1 012 .5/ 8 .8 ,2 0 13 +0
1
gˆ(t) = ,2
.5yt−k + yt−k+1 + · · · + yt+k−1 + .5yt+k 2k
5,3 0 k 6= 6 9 13 ./0 gˆ ./ 718.,/ 8 0 0/82.05
0 ,40 ,2
.-720
50
.
,/81./5 8 0
/0 05512 120 5,3 0 ,
+./05
& &
,7+4 8 0 ,++, ./- +./05 ,2 ./58014 & &
,2
/,8
.5 1 -,,4 .-720 1/4 8 0 1 ,3 1/ ./- 80 8 57--058 8 18 3 ,40+ ,2 8 0 4181 78 8 18 4,05/ 8 5 7120 Y.8t ≈8 Y0t−1 ,9502 18.,/ 8 18 8 0 5 1 0 1 -0/021++ ./ 2015./- 820/4 Yt
1 ./4 1 71/8.818. 0 1 8, 5 , 8 0 820/4 9 1 8 0 1/1+ 5.5 3 .5504 8 0 820/4 3 2, 0 8 0 1/1+ 5.5 5, .8 5 ,/5.580/8 .8 8 0 820/4 .-7205 1/4 1/4 8 0 1 ,3 1/ ./- 80 8 1/1+ 0 8 0
8.3 0 502.05 15 8 ,7- .8 15 8 0 513 0 5827 8720 8 2,7- ,78 8 0 0/8.20 8.3 0 ,05 8 18 3 1 0 50/50 ./ , 1/4 .3 +03 0/8 5,3 0 1 , ./ 058. -18./- 08 02 8 0 5827 8720 , 8 0 502.05 1/-05 2,3 012+ 8, +180
6 ,,50 ,/0 ,2 3 ,20 , 8 0 ,8 02 8, 12 085 8 18 ,3 0 .8 8 0 4181 / 058.-180 08 02 .8 15 8 0 513 0 5827 8720 15 8 0
1 9
1 0 1 +175.9+0 1/1+ 5.5 , 8 0 7/- .50150 4181 15 8 0 8 200 4181 5085 1/4 . 120 8 0 8,81+ 4018 5 8 0 4018 5 , 03 1+05 1/4 8 0 4018 5 , 3 1+05 , 8 0 4018 5 , 03 1+05 1/4 3 1+05 ,++, 5.3 .+12 4.582.978.,/1+ 18802/5 758. ,72 1/5 02
1 0 1 +175.9+0 1/1+ 5.5 , 8 0 205.40/85 1 2, 1+ 218./-5 1 9
1 0 1 +175.9+0 1/1+ 5.5 , 8 0 42. 025 4018 5
,24./- 8, 6,3 7+5,2 012./- , 5018 90+85 15 ./82,47 04 ,/ 1/ .4 8 18 0 0 8 8 0 /73902 , 4018 5 758. ,72 1/5 02 5 8 0 /73902 , 4018 5 20+1804 8, 8 0 /739 02 , .+,3 08025 42. 0/ 50 8 0 12.19+0 ./ 8 0 4181 508 758. ,72 1/5 02
@ = A &=#$& * ;# #$*#$ *
!
6,/5.402 8 0 ,++, ./- 8 , 1 85
08 . . 4 ,. 6 1 802 02 .50 5 , 04 8 18 Y1 , . . . , Yn ∼ P `(λ) 40 0/45 ,/+ ,/ 1/4(λ) /,8 ,/ 8 0 5 0 . 1+705 , 8 0 ./4. .471+ Yi 5
Yi
08 . . 4 6 1 802 02 .50 5 , 04 8 18 Y1 , . . . , Yn ∼ P `(λ) 40 0/45 ,/+ ,/ 1/4(λ) /,8 ,/ 8 0 5 0 . 1+705 , 8 0 ./4. .471+ Yi 5
Yi
728 02 5./ 0 71/8. 05 , 582,/-+ 8 0 4181 57 ,28 01 1+70 , `(λ) ,8 02 15 0 85 , 120 .220+0 1/8 ,2 ./ 020/ 0 19 ,78 .8 57 05 8, /,λ y λ 1/4 8 020 ,20 .8 57 05 8, /, P 0 4,/ 8 /004 8, /, 8 0 `(λ) i ./4. .471+ 5 0 51 8 18 P .5 1 57 Y.0/8 5818.58. ,2 Yi Yi λ 0 8.,/ 0 13 ./05 8 0 -0/021+ ,/ 0 8 , 57 .0/ 0 ,2 ./ 8 0 ,/80 8 , 1 1213 082. 13 .+ 0 .401 , 57 .0/ .5 ,23 1+. 04 ./ 0 /.8.,/
9 1 1213 0802 7/ /, / 08 θ
15
θ
08
90 1 13 .+ , 2,919.+.8 40/5.8.05 ./40 04
08{p(· | θ)} 9 0 1 513 +0 2,3 ,2 5,3 0 (y1 , . . . , yn ) p(· | θ) 9 y0 1=5818.58. 57 8 18 8 0 ,./8 4.582.978.,/ 1 8,25 T (y) Y p(yi | θ) = g(T (y), θ)h(y).
5,3 0 7/ 8.,/5 1/4
,2
g
0
h
0/
T
.5 1++04 1
θ
.401 .5 8 18 ,/ 0 8 0 4181 1 0 900/ ,9502 04 .5 1 ,/581/8 h(y) Q 8 18 4,05 /,8 40 0/4 , 5, θ `(θ) ∝ p(y | θ) = g(T, θ)h(y) ∝ g(T, θ) 020 ,20 ./ ,2402 8, /, 8 0 +. 0+. ,,4 7/ i8.,/ 1/4 3 1 0 ./ 020/ 0 19,78 0 ,/+ /004 8, /, /,8 1/ 8 ./- 0+50 19 ,78 ,2 ,72 ,.55,/ θ T (y) y P 1/4 ,/0/8.1+ 0 13 +05 0 1/ 81 0 T (y) = yi ,2 1 3 ,20 4081.+04 +,, 18 57 .0/ 8 ./ , -0/0218./8 200 02/ (θ) 82.1+5 1/ 90 -0/021804 ,9 .,75+ 9 -0/0218./y ≡ (y1 , y2, y3 ) y y1 , y2 , y3 50 70/8.1++ 0 ,55.9+0 ,78 ,3 05 1/4 8 0.2 2,919.+.8.05 120
(0, 0, 0)
(1 − θ)3
(1, 0, 0) (0, 1, 0) θ(1 − θ)2 (0, 0, 1) (1, 1, 0) (1, 0, 1) θ2 (1 − θ) (0, 1, 1) 78
y
1/
θ3
(1, 1, 1)
1+5, 9 0 -0/021804 9 1 8 , 580 2, 04720
0/02180 P .8 2,919.+.8.05 y = 0, 1, 2, 3 3 205 0 8.i 0+
θ) θ
1
9
P
yi = 0
P
yi = 1
P
yi = 3
19.+.8
P
8
(1, 0, 0) (0, 1, 0)
19.+.8yi =2
4
-0/02180 (0, 0, 0) -0/02180
(1−θ)3 3θ(1−θ)2 3θ2 (1−
-0/02180
-0/02180
(1, 1, 0) (1, 0, 1) (1, 1, 1)
,2
,2
(0, 0, 1) (0, 1, 1)
01 .8 2,9 01 .8 2,9
.5 015 8, 0 8 18 8 0 8 , 580 2, 04720 -0/021805 01 , 8 0 ,55.9+0 ,78 ,3 05 .8 8 0 513 0 2,919.+.8.05 15 8 0 ,9 .,75 50 70/8.1+ 2, 04720 ,2 -0/0218./8 0 8 , 2, 047205 120 0 7. 1+0/8 78 ./ 8 0 8 , 580 y 2, 04720 ,/+ 8 0 258 580 40 0/45 ,/ , . 0 1/8 8, 750 8 0 4181 8, θ
+012/ 19,78 0 ,/+ /004 /, 8 0 ,78 ,3 0 , 8 0 258 580 0 50 ,/4 580 .5 .220+0θ 1/8 0 0 ,/+ /004 8, /, P / ,8 02 ,245 P .5 yi yi 57 .0/8 ,2 1/ 0 13 +0 , 1/,8 02 8 0 +08 . . 4 18 .5 1 y1 , . . . , yn ∼ (0, θ) 57 .0/8 5818.58. ( . ,2 1 yi < θ i = 1, . . . , n θn p(y | θ) = ,8 02 .50 5 , 5 8 18 5818.58.
0 1 = n 1(0,θ) (y(n) ) θ 8 0 3 1 .373 , 8 0 5 y(n) yi
.5 1 ,/0 4.3 0/5.,/1+ 57 .0/8
B I I I IF I I N I I I I IF I T I IF D I IF I II I IF E F I IF x1 , . . . , xn I I I I I I (n) IF x
I I I I I ,3 08.3 05 57 .0/8 . . 4 13 0/
x(n)
5818.58. 5 120 .- 02 4.3 0/5.,/1+ 08
(α, β)
Y
5,
p(yi | α, β) =
Y
1 y α−1 e−yi /β = Γ(α)β α i
1 Γ(α)β α
y1 , . . . , yn ∼
n Y P α−1 yi e− yi /β
.5 1 8 , 4.3 0/5.,/1+ 57 .0/8 5818.58. .5 1 57 .0/8 5818.58. 7 .0/8 5818.58. 5 120 /,8 7/. 70 T = T (y) 1/4 . .5 1 7/ 8.,/ 8 0/ .5 1+5, 57 .0/8 , ./ 8 0 ,.55,/ f f (T ) P ,/0/8.1+ 1/4 02/,7++. 0 13 +05 020 P 15 57 .0/8 yi y¯ = /n .5 1+5, 57 .0/8 78 8 0 +1 , 7/. 70/055 .5 0 0/ 3 ,20 50 020 0 y i,+0 4181 508 .5 1/ 4.3 0/5.,/1+ 57 .0/8 5818.58. 90 1750 Q P T (y) = ( yi , yi )
T (y) = (y)
020
n Y p(yi | θ) = g(T (y), θ)h(y)
1/4
+5, 8 0 5818.58. +5, .
g(T (y), θ) = p(y | θ) h(y) = 1 .5 1/,8 02 4.3 0/5.,/1+ 57 .0/8 n (y(1) , . . . , y(n) )
T (y) = .5 1/ T
.5 1 8 , 4.3 0/5.,/1+ 57 .0/8 ,/0 4.3 0/5.,/1+ 5818.58. 8 0/ 2 = (y1 , T ) 57 .0/8 5818.58. 78 .8 .5 ./87.8. 0+ T+012 8 18 8 050 57 .0/8 5818.58. 5 120 .- 02 4.3 0/5.,/1+ 8 1/ /0 05512 0 1/ 90 2047 04 8, +, 02 4.3 0/ 5.,/1+ 5818.58. 5 .+0 2081././- 57 .0/ 8 18 .5 .8 ,78 +,5./- ./ ,23 1 8.,/ 0 0 .401 ./ 8 0 20 04./- 121-21 .5 8 18 8 0 .- 4.3 0/5.,/1+ 57 .0/8 5818.58. 5 1/ 90 821/5 ,23 04 ./8, 8 0 +, 4.3 0/5.,/1+ ,/05 978 /,8 - .5 1 7/ 8.,/ , 978 .5 y¯ (y(1) , . . . , y(n) ) (y(1) , . . . , y(n) ) /,8 1 7/ 8.,/ , 0 /.8.,/ .5 ,2 5818.58. 5 8 18 1 0 900/ 2047 04 15 37 15 ,55.9+0 y¯ .8 ,78 +,5./- 57 .0/ .5 1++04 . ,2 57 .0/8 5818.58. T (y) 0 02 ,8 02 57 .0/8 5818.58. .5 1 7/ 8.,/ , T2 T (y)
T2 (y)
.5 9,, 4,05 /,8 40+ 0 ./8, 3 08 ,45 ,2 /4./- 3 ./.3 1+ 57 .0/8 5818.58. 5 / 3 ,58 1505 8 0 7502 1/ 20 ,-/. 0 08 02 1 5818.58. .5 3 ./.3 1+ 57 .0/8 ,05 8 0 8 0,2 , 57 .0/ .3 + 8 18 5818.58. .1/5 /004 +,, ,/+ 18 57 .0/8 5818.58. 5 1/4 /,8 18 ,8 02 15 0 85 , 8 0 4181 ,8 7.80 08 9 0 9./12 21/4,3 12.19+05 1/4 57 ,50 0 14, 8 8 0 3 ,40+ y1 , . . . , yn . . 4 02/ 0/ ,2 058.3 18./- 0 /004 +,, ,/+ 18 P y1 , . . . , yn ∼ (θ) θ yi 78 57 ,50 872/ ,78 8, 90
(y1 , . . . , yn )
· · · 0} |1 1 {z · · · 1}, |0 0 {z
. 0 3 1/ 5 ,++, 04 9 3 1/ 5 7 1 4181508 ,7+4 158 4,798 ,/ 8 0 15573 8.,/ 8 18 8 0 5 120 ./40 0/40/8 74-./- 2,3 8 .5 4181508 .8 +,, 5 yi 37 3 ,20 +. 0+ 8 18 8 0 5 ,3 0 ./ 58201 5 , 5818.58. .1/5 5 ,7+4 +,, yi 18 1++ 8 0 4181 /,8 758 57 .0/8 5818.58. 5 90 1750 +,, ./- 18 1++ 8 0 4181 1/ 0+ 75 20180 1/4 2.8. 70 3 ,40+5 78 ,/ 0 1 3 ,40+ 15 9 00/ 14, 804 8 0/ ./ 020/ 0 5 ,7+4 90 91504 ,/ 57 .0/8 5818.58. 5
!
072.58. 1++ 5 01 ./- 15 0 ,++0 8 0 02 3 ,20 4181 0 5 ,7+4 90 19+0 8, +012/ 8 0 8278 0 02 3 ,20 1 72180+ .5 072.58. .5 1 87204 ,23 1++ 18 +0158 ,2 1213 0802 058.3 18.,/ 9 8 0 /,8.,/ , " # $ % , 51 08 02 1/ 058.3 18,2 .5 ,/5.580/8 0 1 0 8, 40 /0 .8 ,2 0 02 513 +0
5. 0 , 8 18 0/4 +08 . . 4 ,2 5,3 0 7/ /, / 40/5.8 1 ./·· ∼ f /.80 3 01/ 1/4 Y1 , Y 2 ,·,2 01 f +08 0 .5 1 n µ σ n ∈ N Tn : R → R Tn 201+ 1+704 7/ 8.,/ , ,2 0 13 +0 . 0 20 82 ./- 8, 058.3 180 (y1 ,P . . . , yn ) 0 3 .- 8 81 0 n −1
µ
Tn = n
1
yi
C 0 50 70/ 0 , " $ $ . ,2 0 02
θ
θ
.5 51.4 8, 9 0 058.3 18,25 T1 , T2 , . . . 1/4 ,2 0 02
" # $ $
>0
lim P[|Tn − θ| < ] = 1.
n→∞
,2 0 13 +0 8 0 1 , 12-0 739025 0,203 51 5 8 0 50 70/ 0 Pn , 513 +0 3 01/5 .5 ,/5.580/8 ,2 .3 .+12+ +08 −1 {T = n y } µ n 1 i P .5 ,/5.580/8 ,2Sn = −1 2 90 8 0 513 +0 12.1/ 0 0/ N {Sn } σ2 (yi − Tn ) i ,20 -0/021++ 3 + 0 5 120 ,/5.580/8
# # $ ˆ $ " B $ Y1 , Y2 , · · · ∼ pY (y | θ) θn $ $ $ " $# " $# " "
(y1 , . . . , yn ) g θ $ $ " #$% " #$# "
# " # $ $ " $# $ " ˆ {g(θn )} "
g(θ)
"" 0 2,, 20 7.205 20-7+12.8 ,/4.8.,/5 20+18./- 8, 4. 020/8.19.+.8 1/4 8 0 ./802 1/-0 , ./80-21+ 1/4 402. 18. 0 8 .5 90 ,/4 8 0 5 , 0 , 8 .5 9 ,,
6,/5.580/ .5 1 -,,4 2, 028 ,/0 5 ,7+4 90 12 , 1/ ./ ,/5.580/8 058.3 18,2 / 8 0 ,8 02 1/4 ,/5.580/ 1+,/0 4,05 /,8 -7121/800 8 18 1 50 70/ 0 , 058.3 18,25 .5 , 8.3 1+ ,2 0 0/ 50/5.9+0 ,2 0 13 +0 +08 8 0 3 01/ , 8 0 258 1+ , Rn (y1 , . . . , yn ) = (bn/2c)−1 (y1 + · · · + y bn/2c ) 8 0 ,9502 18.,/5 .5 8 0 "" , 8 0 +12-058 ./80-02 /,8 -201802 8 1/ bwc w 0 50 70/ 0 .5 ,/5.580/8 ,2 978 .5 /,8 15 -,,4 15 8 0 50 70/ 0 w {Rn } µ , 513 +0 3 01/5 8 5003 5 /18721+ 8, 1/8 8 0 513 +./- 4.582.978.,/ , 1/ 058.3 18,2 8, 90 0/80204 12,7/4 8 0 1213 0802 90./- 058.3 1804 .5 405.4021873 .5 1 87204 ,23 1++ 18 +0158 ,2 0/802./- ./ 8 0 50/50 , 0 0 818.,/ 9 8 0 /,8.,/ , #
G
71/8.8
ˆ −θ E[θ]
#
08 ˆ ˆ 90 1/ 058.3 18,2 , 1 1213 0802 0 θ = θ(y1, . . . , yn ) .5 1++04 8 0 # , ˆ / 058.3 18,2 ,50 9.15 .5 .5θ 1++04
θ
020 120 5,3 0 0 13 +05 . . 4 1/4 ,/5.402 B 08 y1 , . . . , yn ∼ (µ, σ) µ ˆ = y¯ 15 1/ 058.3 180 , 0 1750 .5 1/ 7/9.1504 058.3 180 ,
B
µ
08
E[¯ y ] = µ y¯
y1 , . . . , yn ∼ σ ˆ 2 = n−1
15 1/ 058.3 180 , E[n−1
σ2
. . 4
X
(µ, σ)
µ
1/4 ,/5.402
(yi − y¯)2
X X (yi − y¯)2 ] = n−1 E[ (yi − µ + µ − y¯)2 ] n X X = n−1 E[ (yi − µ)2 ] + 2E[ (yi − µ)(µ − y¯)] o X + E[ (µ − y¯)2 ] = n−1 nσ 2 − 2σ 2 + σ 2 = σ 2 − n−1 σ 2 n−1 2 σ = n
020 ,20 2 .5 1 5818.58. .1/5σˆ 20 02 2
y¯)
9.1504 058.3 18,2 , 2 8 5 9.15 .5 2 ,3 0 σ −σ /n P 8, 750 8 0 7/9.1504 058.3 18,2 2 −1 σ ˜ = (n−1)
(yi −
. . 4 1/4 ,/5.402 ˆ B 08 x1 , . . . , xn ∼ (0, θ) θ = x(n) 15 1/ 058.3 180 , ˆ .5 8 0 3 + 0 500 0 8.,/ 78 θ θ x(n) < θ 8 020 ,20 8 020 ,20 .5 1 9.1504 058.3 18,2 ,
E[x(n) ] < θ
x(n)
θ
!
(= * = K #$ *
@ = K &#&= *
7. 1+0/8 1213 0802. 18.,/5 05 0 .1++ 5 08 / 12.1/ 0 , 5
!"
= K #$
!:
=& ' A #? & *$*
B
B
(& *#$ %
!>
A & #$ K $ $& *
!
#$ L
; &
K $ $& *
, 18.,/ 5 1+0 13 .+.05
< #$ *
7/ 8.,/1+5
=$ &
/ 12.1/ 0
*' K A # #$ *
/ 201+ +. 0 4181 5085 120 /.80 08 0 , 80/ 1 01+ 8, 8 0 1 (y1 , . . . , yn ) , 12-0 739025 ,2 8 0 60/821+ .3 .8 0,203 0,203 5 1/4 . ,/ 02/ 8 0 +.3 .8 , 1 50 70/ 0 , 21/4,3 12.19+05 15 n→∞ 0 , 0 .5 8 18 0/ .5 +12-0 8 ,50 8 0,203 5 .++ 80++ 75 5,3 08 ./- n 18 +0158 1 2, .3 180+ 19,78 8 0 4.582.978.,/ , 8 0 513 +0 3 01/ 78
0 20 1 04 .8 8 0 7058.,/5 , +12-0 .5 +12-0 1/4 , +,50 .5 8 0 1 2, .3 18.,/ , 81 0 1/ 0 13 +0 0 3 .- 8 1/8 8, 1 + 8 0 1 , 12-0 739 025 ,2 8 0 60/821+ .3 .8 0,203 8, 1 50 70/ 0 , 21/4,3 12.19+05 1 , Y2 , . . . 2,3 1 4.582.978.,/ .8 3 01/ 1/4 Y020 120 1 0 ./581/ 05 , 8 0 µ σ 258 50 021+ 0+03 0/85 , 57 1 50 70/ 0
··· ··· · · ·
1
50 70/ 0 , 7 .05 ,/0 2, , 8 0 1221 0 ./4. 1805 8 18 8 0 ··· 50 70/ 0 ,/8./705 ./ /.80+ 0 ./4. 1805 8 18 8 020 120 ./ /.80+ 3 1/ 57 50 70/ 05 0 /739025 020 -0/021804 9 •
•
,50 8, -0/02180 5 2,3 8 0 4.582.978.,/ 5, 7504 Yi (0, 1) 1/4 5, ,2 8 .5 0 13 +0 1/4 ,50 120 129.8212 ,. 05 µ=0 σ=1 ,7+4 1 0 7504 1/ 1+705 , 1/4 1/4 1/ 4.582.978.,/ ,2 . µ σ /, , 8, -0/02180 21/4,3 12.19+05 ,/ 8 0 ,3 7802
4,05 2,7/4./- / 8 .5 150 0 20 2./8./- 01 /73902 .8 8 , 40 .3 1+ +1 05
0 1750 8 020 120 37+8. +0 50 70/ 05 01 .8 37+8. +0 0+03 0/85 0 /004 8 , 5795 2. 85 8, 00 821 , 8 ./-5 2, 02+ 08 90 8 0 8 0+03 0/8 , j ij 8 0 8 50 70/ 0 ,2 8 0 8 50 70/ 0 , 21/4,3 Y12.19+05 0 20 ./80205804 i 020 /4 ./ 8 i 0 50 70/ 0 , 3 01/5 Y¯i1 , Y¯i2, . . . Y¯in = (Yi1 + · · · + √ Yin )/n 0 20 1+5, ./80205804 ./ 8 0 50 70/ 0 020 Zi1 , Zi2 , . . . Z = n(Y¯ − µ) ,2 8 0 8 200 ./581/ 05 19, 0 8 0 5 1/4 5 1/ 9 0 in2./804 .8 in Y¯in Zin
!
,3 7805 1 737+18. 0 573 5, .0+45 8 0 0 8,2 2./8 ,78 . ,7 20 /,8 5720 18 .8 .5 020 ,20 .5 8 0 50 70/ 0 , Y¯in 5 ,3 7805 8 0 5 7120 2,,8 , 8 0 50 ,/4 581803 0/8 2./85 • ! 8 0 50 70/ 0 , 5 •
0
Zin
2057+85 ,2 8 0 ¯ 5 120 Yin
1/4 ,2 8 0
··· ··· · · ·
··· ··· · · ·
5 120
Zin
20 ./80205804 ./ 8 0 ,++, ./- 7058.,/5 .++ 0 02 50 70/ 0 , ¯ 5 ,2 i 5 ,/ 02-0 .5 8 0 +.3 .8 1+,/01 2, Yi, 8 0Z1221
0
8 0 ,/ 02-0
.5 1 7058.,/ 19 ,78
4, 8 0 1++ 1 0 8 0 513 0 +.3 .8
/,8 0 02 50 70/ 0 ,/ 02-05 18 21 8.,/ , 8 03 ,/ 02-0 ,2 5 ,2 5 18 .5 8 0 2,919.+.8 8 18 1 21/4,3 + ,50/ 50 70/ 0 , Y¯i Zi ,/ 02-05 ,2
1 04 18 .5 8 0 4.582.978.,/ , ¯ ,2 n 19,78 8 0 4.582.978.,/ 1+,/- ,+73 /5 , 8 Y0n 1221Zn
.5
.5 1 7058.,/
,05 8 0
4.582.978.,/ , ¯ ,2 40 0/4 ,/ Yn Zn n 5 8 020 1 +.3 .8./- 4.582.978.,/ 15 n→∞ ,3 0 5.3 +0 0 13 +05 1/4 8 0 82,/- 1 , 12-0 739025 0, 203 1/5 02 7058.,/5 1/4 ,2 8 0 50 70/ 05 , ¯ 5 0 60/821+ Y 5 .3 .8 0,203 0,203 1/5 025 7058.,/ ,2 8 0 50i 70/ 05 ,
Zi
.++ 0 02
.5
50 70/ 0 , ¯ 5 ,/ 02-0 , 7 ,50 8 0 50 70/ 0 , 5 Yi Yi 0/ ¯ ./ 201505 .8 ,78 +.3 .8 1/4 4,05 /,8 ,/ 02-0 {Y i }
1, 2, 3, . . .
8 0 ,/ 02-0 4, 8 0 1 0 8 0 513 0 +.3 .8 50 70/ 05 , 5
020 120 8 ,
Yi
0 ,2205 ,/4./-
,
··· ···
50 70/ 05 ¯ ,/ 02-0 8, 4. 020/8 +.3 .85 {Y i } 18 .5 8 0 2,919.+.8 , ,/ 02-0/ 0 0 2,919.+.8 , ,/ 02-0/ 0 .5 18 5 8 0 82,/- 1 , 12-0 739 025 / 128. 7+12 8 0 2,919.+.8 , 21/4,3 + -088./- 1 50 70/ 0 8 18 4,05/ 8 ,/ 02-0 +. 0 .5 78 8 0 82,/- 1 , 12-0 739025 51 5 0 0/ 3 ,20 1, 2, 3, . . . 8 51 5 P[ lim Y¯n = µ] = 1.
, 8 0 2,919.+.8 , -088./8 18 ,/ 02-0 8, 5,3 08 ./-
n→∞
50 70/ 05 57 15 ,2 1, 1, 1, . . . −1, −1, −1, . . . ,8 02 8 1/ .5 µ
0 1//,8 51 ./ -0/021+ 8 40 0/45 18 .5 8 0 4.582.978.,/ , Zn ,/ 8 0 4.582.978.,/ , 8 0 ./4.
.471+ 5 Yi ,05 8 0 4.582.978.,/ , 40 0/4 ,/ 05 0 0 8 ./ 8 0 5 0 .1+ Zn n 150 020 ,2 1++
Yi ∼
(0, 1)
i
5 8 020 1 +.3 .8./- 4.582.978.,/ 05 18 5 8 0 60/821+ .3 .8 0, 203 0-124+055 , 8 0 4.582.978.,/ , 8 0 5 15 +,/- 15 Y Var(Yi ) < ∞ 8 0 +.3 .8 15 , 8 0 4.582.978.,/ , i .5 n→∞
Zn
(0, 1)
0 1 , 12-0 739 025 1/4 8 0 60/821+ .3 .8 0,203 120 8 0,203 5 19,78 8 0 +.3 .8 15 0/ 0 750 8 ,50 8 0,203 5 ./ 21 8. 0 0 n → ∞ , 0 8 18 ,72 513 +0 5. 0 .5 +12-0 0/,7- 8 18 ¯ 1/4 n Zin ∼ (0, 1) in ≈ µ 1 2, .3 180+ 78 , +12-0 5 ,7+4 90 9 0 ,20 Y20+ ./- ,/ 8 050 8 0,203 5 1/4 , -,,4 .5 8 0 1 2, .3 18.,/ n 0 1/5 02 .5 $ " $ # $ # $# " " $ 18 5 18 0 +,, 18 /0 8 Y , .++7582180 0 i-0/02180 50 70/ 05 , 5 2,3 8 , 4.582.978.,/5 ,3 780 5 1/4 5 ,2 50 021+ 1+705 , Y ij1/4 ,3 120 /0 4.582.978.,/ ¯ Yin Zin n .5 8 0 ,8 02 .5 1 20 0/80204 1/4 205 1+04 025.,/ , 0
(0, 1)
(.39, .01)
0 0 40/5.8 5 , / ./ .-720 15 ,50/ ,2 .85 15 3 3 0 .01) 82 8 15 1(.39, 3 01/ , 1/4 1 12.1/ 0 , .39/.40 = .975 (.39)(.01)/((.40)2(1.40)) ≈ 8 15 20 0/80204 1/4 205 1+04 8, 1 0 1 3 01/ , 1/4 12.1/ 0 , .017 .5 8 0 513 0 15 8 0 4.582.978.,/ 1/12 (0, 1) 0/5.8.05 , 8 0 ¯ 5 120 ./ .-720 5 8 0 513 +0 5. 0 ./ 201505 2,3 8, Yin 8 0 ¯ 5 2,3 9 ,8 4.582.978.,/5 -08 +,502 8, 10 n = 270 Y 8 0.2 0n =0 804
1+70 , 18 5in8 0 1 , 12-0 739025 18 ,2 0 13 ,7/8 9 . 8 0 20 , 8 0.2 3 01/ -,05 2,3 19,78 8, 19 ,78 ±.2 ±.04 ,2 /4 /1++ 15 18 5 6,2,++12 18 8 0 40/5.8.05 -08 3 ,20 n→∞ ,23 1+ 18 5 8 0 60/821+ .3 .8 0,203 18 ,2 ,80 8 18 8 0 40/5.8 , 8 0 5 402. 04 2,3 8 0 4.582.978.,/ .5 Y¯in +,50 8, ,23 1+ 0 0/ ,2 8 0 53 1++058 513 +0 5. 0 .+0(0,8 1)0 40/5.8 , 8 0 5 402. 04 2,3 8 0 0 4.582.978.,/ .5 1 , 18 5 9 0 1750 Y¯in (.39, .01) .5 5 3 3 082. 1/4 7/.3 ,41+ 1/4 8 020 ,20 +,50 8, ,23 1+ 8, 90-./ (0, 1) .8 .+0 0 .5 12 2,3 5 3 3 082. 1/4 7/.3 ,41+ 1/4 8 020 ,20 (.39, .01) 12 2,3 ,23 1+ 8, 9 0-./ .8 , 0 /0045 1 +12-02 8, 3 1 0 8 0 (.39, .01) 1 -,,4 1 2, .3 18.,/n 60/821+ .3 .8 0,203 ,2 . 0 8, 90 .-720 .5 ,2 8 0 5 8 5 8 0 513 0 15 .-720 0 0 8 8 18 01 Zin 40/5.8 15 900/ 20 0/80204 1/4 205 1+04 8, 1 0 3 01/ 1/4 12.1/ 0 0/ 78 ,/ 8 0 513 0 5 1+0 0 1/ 500 8 18 1++ 40/5.8.05 120 ,/ 02-./- 8,
4
6
8
2
(0, 1)
0
0.0
0.2
0.4
.-720 0 0
0.6 (.39, .01)
0.8
40/5.8
1.0
.-720 15 2,47 04
9
! .-720 15
-0/021804 9 8 0 ,++, ./-
,40
! ! ! •
25 20 15 10 5 0
0
5
10
15
20
25
30
n = 30
30
n = 10
0.0
0.4
0.8
0.0
0.8
25 20 15 10 5 0
0
5
10
15
20
25
30
n = 270
30
n = 90
0.4
0.0
0.4
0.8
0.0
0.4
0.8
Y¯in (0, 1) (.39, .01)
Y1 , . . . , Yn
µ µ
σ
σ
(µ, σ)
µ σ
Y1 , . . . , Yn (α, β)
Y1 , . . . , Yn
∼
(−θ, θ)
0.8 0.6 0.4 0.2 0.0
0.0
0.2
0.4
0.6
0.8
1.0
n = 30
1.0
n = 10
−3
−1
1
3
−3
1
3
0.8 0.6 0.4 0.2 0.0
0.0
0.2
0.4
0.6
0.8
1.0
n = 270
1.0
n = 90
−1
−3
−1
1
3
−3
−1
1
3
Zin (0, 1) (.39, .01)
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