Prml_excercise 2.8
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PRML 2 :2.8
Odj
M-1 7
!W$S
n 18 |
1/4
dj 2.8 DNN(Qt x, y =7F=liN1~N(,[ p(x, y) ,,+kH9 k%J
2
E[x] = Ey [Ex [x|y]]
(2.270)
var[x] = Ey [varx [x|y]] + vary [Ex [x|y]]
(2.271)
0Nf E [x|y]'roU-N(,[ p(x|y) NlgN x N|TM$ var [x|y]'roU-N(,[ p(x|y) NlgN x N,6 x
x
2/4
0 (2.270) NZ@ Z@'
E[x] = Ey [Ex [x|y]]
(2.270)
jA Ey [Ex [x|y]] ≡ xp(x|y) dx p(y) dy Z Z = p(x|y)p(y)dy x dx Z = p(x)x dx Z Z
jA
≡ E[x]
3/4
0 (2.271) NZ@ var[x] = Ey [varx [x|y]] + vary [Ex [x|y]]
(2.271)
Z@'gK,6N-AG"k'var [z] = E [z ] − E [z] r 3 sH& 3H% z
z
2
z
2
var[x] = Ex [x2 ] − Ex [x]2 2 0 (2.270) 2 = Ey Ex [x |y] − Ey Ex [x|y]
(1)
∵ Ex [x2 |y] = varx [x|y] + Ex [x|y]2 (1)
0 Ke~9kH$ 0 1 = E var [x|y]+ E [x|y] − E E [x|y]
2
2
y
x
x
y
2
x
= Ey varx [x|y] + Ey Ex [x|y] − Ey Ex [x|y] = Ey varx [x|y] + vary Ex [x|y]
2 4/4
Odj
M-1 7
!W$S
n 18 |
1/4
dj 2.8 DNN(Qt x, y =7F=liN1~N(,[ p(x, y) ,,+kH9 k%J
2
E[x] = Ey [Ex [x|y]]
(2.270)
var[x] = Ey [varx [x|y]] + vary [Ex [x|y]]
(2.271)
0Nf E [x|y]'roU-N(,[ p(x|y) NlgN x N|TM$ var [x|y]'roU-N(,[ p(x|y) NlgN x N,6 x
x
2/4
0 (2.270) NZ@ Z@'
E[x] = Ey [Ex [x|y]]
(2.270)
jA Ey [Ex [x|y]] ≡ xp(x|y) dx p(y) dy Z Z = p(x|y)p(y)dy x dx Z = p(x)x dx Z Z
jA
≡ E[x]
3/4
0 (2.271) NZ@ var[x] = Ey [varx [x|y]] + vary [Ex [x|y]]
(2.271)
Z@'gK,6N-AG"k'var [z] = E [z ] − E [z] r 3 sH& 3H% z
z
2
z
2
var[x] = Ex [x2 ] − Ex [x]2 2 0 (2.270) 2 = Ey Ex [x |y] − Ey Ex [x|y]
(1)
∵ Ex [x2 |y] = varx [x|y] + Ex [x|y]2 (1)
0 Ke~9kH$ 0 1 = E var [x|y]+ E [x|y] − E E [x|y]
2
2
y
x
x
y
2
x
= Ey varx [x|y] + Ey Ex [x|y] − Ey Ex [x|y] = Ey varx [x|y] + vary Ex [x|y]
2 4/4
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