University of Illinois Fall 2003
ECE 313: Problem Set #13
Assigned: Friday, December 5, 2003
Due: Friday, December 12, 2003
1. Assume that Var(X1+X2+X3) = Var(X1) + Var(X2) + Var(X3). Can we conclude that
X1,X2, and X3are pairwise uncorrelated? Why?
2. Let E[X] = 1, E[Y] = 4, Var(X) = 4, Var(Y) = 9, and ρX,Y = 0.1.
(a) If Z= 2(X+Y)(X−Y), find E[Z].
(b) If T= 2X+Yand U= 2X−Y, find Cov(T, U ).
(c) If W= 3X+Y+ 2, find E[W] and Var(W).
3. Assume that the density function fX(u) of a random variable Xis an even function, and
that E(X2)<∞. Show that (a) |X|and Xare uncorrelated, and (b) |X|and Xare not
independent.
4. Let X1, X2, . . . , Xnbe independent and identically distributed positive RVs. Show that for
k≤n,
E"Pk
i=1 Xi
Pn
i=1 Xi#=k
n.
5. Let X1, X2,··· be an infinite sequence of independent, identically distributed, random
variables with mean µand variance σ2. We define Yn=Xn+Xn+1 +Xn+2, for n= 1,2,···.
For each k≥0, compute Cov(Yn, Yn+k).
6. Suppose that Xand Yare jointly Gaussian random variables. It is further known that:
E[X] = 0, E[Y] = 0,Var(X) = σ2
1,Var(Y) = σ2
2, ρ(X, Y ) = ρ.
Find an angle θsuch that Z=Xcos θ+Ysin θand W=Ycos θ−Xsin θare independent
Gaussian random variables. You may express your answer in terms of a trigonometric
function of σ1, σ2and ρ. In particular, what is the value of θif σ1=σ2?
7. Extra Credit: Show that |Cov(X, Y )|2≤Var(X)Var(Y), where the equality holds if and
only if Y=aX +bwith aand bbeing some arbitrary constants.
8. Extra Credit: Let X1and X2be independent RVs with the same distribution N(µ, σ2).
Show that E[max{X1, X2}] = µ+σ/√π.