Utah State University
ECE 6010
Stochastic Processes
Homework # 3
Due Friday Sept. 23, 2005
1. Suppose Xand Yare the indicator functions of events Aand B, respectively. Find
ρ(X, Y ), and show that Xand Yare independent if and only if ρ(X, Y ) = 0.
2. Suppose φ(u) is a ch.f. Show that |φ(u)|2is also a ch.f.
3. Suppose Xand Yare jointly Gaussian. Use ch.f.s to show that ρ(X, Y ) = ρ.
4. Suppose Xand Yare jointly continuous. (a) Show that
FY|X(b|x) = Zb
−∞
fXY (x, y )
fX(x)dy
and thus that
fY|X(y|x) = fXY (x, y)
fX(x)
(b) Suppose R∞
−∞ |y|fY|X(y|x)dy < ∞. Show that E[Y|X=x] = R∞
−∞ yfY|X(y|x)dy.
5. Suppose Xand Yare independent continuous r.v.s with c.d.f.s FXand FY, respectively.
Suppose further that FX(b)≥FY(b) for all b∈R. Show that P(X≥Y)≤1/2.
6. Prove Jensen’s inequality for the case of simple-function r.v.’s
7. Prove the Schwartz inequality.
Problems from Grimmet & Stirzaker:
1. Prob 2.7.4
2. Prob 2.7.7. Hint: binomial distribution
3. Prob 2.7.9. Hint:
P(X−≤x) = (0x < 0,
1−limy↑−xF(y)x≥0.
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