EE 126, Fall 2001
Midterm #2
Professor Anantharam
The exams starts at 3:40 p.m. sharp and ends at 5:00 p.m. sharp.
There are 5 problems. The maximum score is 50 points.
The exam is open book and notes.
Problem #1 - 25 points
For each of the following statements, indicate whether you believe that the statement is true or believe it is
false, and give a brief explanation of your reasoning. A correct answer without a valid explanation gets 1
points. A correct answer with a valid explanation gets 5 points.
(a) If X is a Gaussian random variable the X, 2X, and 3X are jointly Gaussian random variables.
(b) Let X, Y, and Z be random variables, which you may assume have a joint density. Let W = Y + Z. Then
E[W | X] = E[Y | X] + E[Z | X]
(c) Let X, Y, and Z be random variables, which you may assume have a joint density. Let W = Y + Z. Then
E[X | W] = E[X | Y] + E[X | Z]
(d) If X is Gaussian and Y is uncorrelated with X, then X and Y are independent.
(e) If Gaussian random variables X and Y have the same mean and the same second moment then they have
the same fourth moment.
Problem #2 - 7 points
Let X ~ N(2,2). Let the conditional density of Y given X be given by
fY|X(y|x) = (1/((3/21/2)*(2*PI)1/2))*e(-1/2)*(2/9)*(y-(1/2)*(x-2)-3)2,
i.e. conditional on X, Y is Gaussian with mean (1/2)(X - 2) + 3 and variance 9/2. Find the density of Y.
Problem #3 - 6 points
I throw three darts at a disk of radius R0 centered at the origin. Each dart lands on the disk and the point at
which it lands is distributed according to
fR THETA(r, theta) = { (3*r2/(2*PI*R03) if 0 <= r <=R0 and -PI <= THETA <= PI
0elsewhere
EE 126, Midterm #2, Fall 2001
EE 126, Fall 2001 Midterm #2 Professor Anantharam 1
