ECE 461 Fall 2006
August 24, 2006
HOMEWORK ASSIGNMENT 1
Reading: Text, Chapter 2
Due Date: August 31, 2006 (in class)
1. Use your knowledge of Gaussian and jointly Gaussian pdfs to get the answers to the
following directly (without resorting to integration).
(a) Find the variance of the random variable that has density
fX(x) = 1
√4πe−(x−3)2
4,for all x.
(b) Suppose fX,Y (x, y ) = 1
2πλ2e−x2+y2
2λ2. Find E[X2+Y2].
2. Let X1, X2,...,Xnbe i.i.d. random variables each with pdf fX(x).
(a) Find the pdf of Y= min{X1, X2,...,Xn}.
(b) Find the pdf of Z= max{X1, X2,...,Xn}.
3. Random variables Xand Yare jointly Gaussian with means mX= 1, mY= 2,
variances σ2
X= 4, σ2
Y= 9, and Cov(X, Y ) = −4.
(a) Find the correlation coefficient between Xand Y.
(b) If Z= 2X+Yand W=X−2Y, find Cov(Z, W ).
(c) Find the pdf of Z.
(d) Find the joint pdf of Zand W.
4. Suppose Xis a random variable that is uniformly distributed on the interval [0,1],
that is fX(x) is 1 on the interval [0,1] and 0 otherwise.
(a) Suppose Y=e−2X. Find FY(y) and fY(y).
Hint: Begin with FY(y) = P{Y≤y}= P{e−2X≤y}= P{X≥ −(log y)/2}.
(b) Now suppose that a random process (which is only defined for t > 0) is given by
Y(t) = e−tX . Find the cdf and pdf of the random variable Y(t0), where t0is a
fixed positive number.
c
V.V. Veeravalli, 2006 1
