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INSTRUCTOR’S SOLUTIONS MANUAL
Probability, Statistics, and Random Processes for Electrical Engineering
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Vector Random Variables
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Functions of Several Random Variables
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Expected Values of Vector Random Variables
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If we write out the quadratic form in the exponent we obtain:
n
∑
j =1
n
n −1
k =1
j =1
∑ Q jk ( x j − m j )( xk − mk ) − ∑
n −1
∑Q
k =1
( x j − m j )( xk − mk )
jk
n −1
=Qnn ( xn − mn ) 2 + 2( xn − mn )∑ Q jk ( x j − m j )
j =1
⎧
1
= Qnn ⎨( xn − mn ) 2 + 2( xn − mn )
Qnn
⎩
n −1
∑Q
j =1
jk
⎫
( x j − m j )⎬
⎭
= Qnn {( xn − mn ) 2 + 2( xn − mn ) B + B 2 } − Qnn B 2
=Qnn {( xn − mn ) + B} − Qnn B 2
2
where B =
1
Qnn
n −1
∑Q
j =1
jk
(xj − mj )
In the first line, all the terms in the second summation are contained in the first
summation. We then completed the square to obtain an expression involving xk, its
mean mk – B, and its variance 1/Qkk. The term QkkB2 is part of the normalization
constant. We therefore conclude that:
⎧⎪ 1
exp ⎨−
⎪⎩ 2Qnn
f X n ( xn | x1 ,..., xn −1 ) =
⎛
1
⎜ x − mn +
Qnn
⎝
⎞
Q jk ( x j − m j ) ⎟
∑
j =1
⎠
n −1
2
⎫⎪
⎬
⎪⎭
2π Qnn
We see that the conditional mean of xn is a linear function of the observations x1, x2,…,
xn-1.
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(a)
(b)
The following Octave code generates a sequence of 1000 samples:
X = normal_rnd(0, 1, 1, 1000);
Y = (X + [0 X(1:length(X)-1)])./2;
cov(Y);
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