Utah State University
ECE 6010
Stochastic Processes
Homework # 4
Due Friday Oct. 7, 2005
1. Suppose X∼ N (µ,Σ).
(a) Show that E[X] = µand cov(X,X) = Σ.
(b) Show that AX+b∼ N (Aµ+b, AΣAT).
(c) Suppose Σ >0 and write Σ = C CT. Show that C−1(X−µ)∼ N (0, I).
2. Suppose you have a random number generator which is capable of generating random
numbers distributed as X∼ N (0,1). Describe how to generate random vectors Y∼
N(µ,Σ).
3. Suppose Xand Yare r.v.s. Show that E[(X−h(Y))2] is minimized over all functions
hwhen his the function
h(y) = E[X|Y=y].
Assume E[X2]<∞.
4. Let (X, Y )∼ N (µx, µy, σ 2
x, σ2
y, ρ). Let X= [ X
Y]∼ N(µ,Σ), where
Σ = s11 s12
s21 s22.
Determine the relationship between µx, µy, σ2
x, σ2
y, ρ and µand Σ.
5. Suppose X=hX1
X2
X3i∼ N(µ,Σ) where
µ=
1
2
3
Σ =
4 2 1
2 6 3
1 3 8
(a) The value X1= 1.5 is measured. Determine the best estimate for (X2, X3).
(b) In a separate problem, the values X2= 1 and X3= 5 are measured. Determine
the best estimate of X1.
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