sm8_8

Prévia do material em texto

Problem 8.8 Show that the mean and variance of a Gaussian random variable X with the density 
function given by Eq. (8.48) are µX and . 
2
Xσ
 
Solution 
Consider the difference E[X]-µX: 
 
[ ] ( ) ( )∫∞
∞− ⎭⎬
⎫
⎩⎨
⎧ −−−=− dxxxX
X
X
X
X
X 2
2
2
exp
2 σ
µ
σπ
µµE 
 
Let y = Xx µ− and substitute 
 
[ ]
0
2
exp
2
2
2
=
⎟⎟⎠
⎞⎜⎜⎝
⎛−=− ∫∞∞− dyyyX
XX
X σσπµE 
 
since integrand has odd symmetry. This implies [ ] XXE µ= . With this result 
 
( ) ( )
( ) ( )∫∞∞− ⎭⎬
⎫
⎩⎨
⎧ −−−=
−=
dxxx
xX
X
X
X
X
X
2
22
2
2
exp
2
Var
σ
µ
σπ
µ
µE
 
 
In this case let 
 
X
Xxy σ
µ−= 
 
and making the substitution, we obtain 
 
dyyyX X ⎭⎬
⎫
⎩⎨
⎧−= ∫∞∞− 2exp2)(Var
22
2
πσ 
 
Recalling the integration-by-parts, i.e., ∫ ∫−= vduuvudv , let u = y and 
dyyydv ⎟⎟⎠
⎞
⎜⎜⎝
⎛ −=
2
exp
2
. Then 
 
Continued on next slide 
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only 
to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that 
permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. 
page…8-8 
Problem 8.8 continued 
 
( )
2
2
2
2
2
2
10
2
exp
2
1
2exp2
)(Var
X
X
XX dy
yyyX
σ
σ
πσπσ
=
•+=
⎟⎟⎠
⎞
⎜⎜⎝
⎛−+⎟⎠
⎞⎜⎝
⎛−−= ∫∞∞−
∞
∞−
 
 
where the second integral is one since it is integral of the normalized Gaussian 
probability density. 
 
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only 
to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that 
permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. 
page…8-9