University of Illinois Fall 2003
ECE 313: Problem Set #12
Assigned: Monday, December 1, 2003
Due: Friday, December 5, 2003
1. Suppose X, Y and Zare independent random variables that are each equally likely to be either 1
or 2. Find the pmf of
(a) XY Z
(b) XY +Y Z +XZ
(c) X2+Y2+Z2
2. If Uis uniform on (0,2π) and Z, independent of U, is exponential with parameter λ= 1,
(a) Find the joint density of the random variables X, Y defined by
½X=√2Zcos U
Y=√2Zsin U
(b) Show that Xand Yare independent unit normal RVs, i.e., that they are independent and
each of them is distributed as N(0,1).
(c) What is the pdf of the random variable R=√X2+Y2?
3. X1and X2are independent random variables. X1is uniformly distributed over (−1,1) and X2is
exponentially distributed with parameter λ= 1.
(a) Find the pdfs of the RVs W=X1+X2and Z=X1/X2.
(b) Find the joint pdf of Wand Z.
4. Let (X, Y ) be uniformly distributed on the interior of the square with vertices at (0,0),(0,1),(1,1)
and (1,0).
(a) Are Xand Yindependent? Are they uncorrelated?
(b) Are random variables (X+Y) and (X−Y) uncorrelated or independent?
Now let (X, Y ) be uniformly distributed on the interior of another square with vertices at (2,0),(1,1),
(0,0) and (1,−1).
(b) Determine whether Xand Yare uncorrelated or independent in this case. What does this
say about random variables being independent and uncorrelated?
(c) Are random variables (X+Y) and (X−Y) uncorrelated or independent?
5. Extra Credit: If the random variables A, B, C are independent, and chosen uniformly in the
interval [0,1], what is the probability that all of the roots of the equation Ax2+Bx +C= 0 are
real?