ECE 313 Problem Set # 12 Fall 2002
Assigned: 11/15/02 Due: Wednesday 11/20/02
More jointly distributed random variables
Assigned reading: Ross, Sections 6.1-6.5, 6.7, 5.4.1,and the first three pages of Chapter 7
Noncredit exercises: Chapter 6, Problems 1,10-15,20-23,28-30,41-43,51,54 and Theoretical Ex. 8,14,22,23,33
1. Uniform distribution on a rotated square
Random variables Xand Yhave a uniform joint density on the square bounded with corners at the points:
(1,0),(0,1),(−1,0) and (0,−1).
(a) Calculate the marginal pdfs of Xand Y. Are Xand Yindependent?
(b) Compute E[X] and V ar(X).
(c) Calculate the pdf of the random variable A=X+Y.
(d) Calculate the pdf of the random variable C=X/Y .
2. Functions of independent exponential random variables
Let X1and X2be independent random varibles, with Xibeing exponentially distributed with parameter λi.
(a) Find the pdf of Z= min{X1, X2}.
(b) Find the pdf of R=X1
X2.
3. Sums of independent normal random variables
Ross, Problem 33, page 293.
4. Sizing a Confidence Interval by the normal approximation to the binomial distribution
A campus network engineer would like to estimate the fraction pof packets going over the fiber optic link
from the campus network to Bardeen Hall that are digital video disk (DVD) packets. The engineer writes a
script to examine 1000 packets, counts the number Xthat are DVD packets, and uses ˆp=X
1000 to estimate
p. The inspected packets are separated by hundreds of other packets, so it is reasonable to assume that each
packet is a DVD packet with probability p, independently of the other packets.
(a) If p= 0.5, estimate P[|ˆp−p| ≥ 0.02].
(b) If p= 0.1, estimate P[|ˆp−p| ≥ 0.02]
(c) If p= 0.5, find the number δso that P[|ˆp−p| ≤ δ]≈0.99. Equivalently, P[p∈[ˆp−δ, ˆp+δ]] ≈0.99.
Note that pis not random, but the confidence interval [ˆp−δ, ˆp+δ] is random.
(d) If p= 0.1, find the number δso that P[|ˆp−p| ≤ δ]≈0.99.
(e) However, the campus network engineer doesn’t know pto begin with, so she can’t select the halfwidth
δof the confidence interval as a function of p. A reasonable approach is to select δso that, the normal
approximation to P[p∈[ˆp−δ, ˆp+δ]] is greater than equal to 0.99 for any value of p. What is that value of
δ?
(f) Using the same approch as in part (e), how many observations are needed (not depending on p) so that
the (random) confidence interval [ˆp−0.01,ˆp+ 0.01] contains pwith probability at least 0.99 (according to
the normal approximation of the binomial)?
5. Transformation of jointly continuous random variables, I
Suppose Zis exponentially distributed with parameter λ= 1, Θ is uniform on (0,2π), and Zis independent
of Θ. Let X=√2Zcos Θ and Y=√2Zsin Θ.
(a) Find the joint pdf of Xand Y. In particular, show that Xand Yare independent of each other, and
each has the standard normal distribution.
(b) Find the pdf of the random variable R=√X2+Y2. What is the name of this distribution?
6. Transformation of jointly continuous random variables, II
Suppose (U, V ) has joint pdf
fU,V (u, v) = 9u2v2if 0 ≤u≤1 & 0 ≤v≤1
0 else
Let X= 3Uand Y=UV . Find the joint pdf of Xand Y, being sure to specify where the joint pdf is zero.
(b) Using the joint pdf of Xand Y, find the conditional pdf, fY|X(y|x), of Ygiven X. (Be sure to indicate
