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University of Illinois at Urbana-Champaign Department of Electrical and Computer Engineering ECE 434: Random Processes Spring 2004 Midsemester Exam 1
Wednesday, March 17, 5:00–7:00pm, 165 Everitt Laboratory
READ THESE COMMENTS BEFORE STARTING THE EXAM!
- This is a closed-book exam! You are allowed one sheet of handwritten notes (both sides). Calculators should not be necessary, but feel free to use one.
- Write your name on the answer booklet.
- There are five unequally weighted problems for a total of 50 points. A bonus problem worth 5 points is also included. Problems are not necessarily in order of difficulty.
- A correct answer does not guarantee credit; an incorrect answer does not guarantee loss of credit. Provide clear explanations, show all relevant work and justify your answers! If we cannot make sense of your writing or reasoning, you may loose points.
- Read each problem carefully and think before performing detailed calculations.
- Only the supplied answer booklet is to be handed in. No additional pages will be considered in the grading. You may want to work things through in the blank areas of the exam and then neatly transfer to the answer sheet the work you would like us to look at.
Problem 1 (12/50, equally weighted parts)
This problem has six independent true/false questions.
(a) If X and Y are zero-mean random variables, then the linear minimum mean squared error (LMMSE) estimator of X based on the observation Y = y is of the form X̂ (y) = αy for some appropriate constant α.
(b) Random variable X is Gaussian and so is random variable Y. Then, X and Y are independent if they are uncorrelated.
(c) Let X be a random variable in the interval [− 1 , 1]; its cdf is unknown but we know that E[X^2 ] = 18. Then, it is possible that P (X ≥ 12 ) = 34.
(d) Random variable X is to be estimated based on observations of the random variable Y via a nonlinear estimator of the form X̂ (y) = αy^2. The constant α is chosen so that E[(X − αY 2 )^2 ] is minimized; then, the following equality is necessarily true E[XY 2 ] = αE[Y 4 ].
(e) Let X be a random variable with a known distribution that is symmetric about x =
- Given the observation that the random variable Y = X^2 takes the value y, the minimum mean squared error (MMSE) estimator X̂ M M SE (y) results in mean squared error E[(X − X̂ M M SE (Y ))^2 ] = cov(X).
(f) Let X 1 , X 2 , ... be a sequence of independent (but not identical) random variables with E[Xi] = 0 and cov(Xi) = 1 for all i. Then, the sequence of random variables
Yn =
∑n i √=1 Xi n converges in distribution to a Gaussian random variable X with zero mean and unit variance.
Problem 3 (12/50, equally weighted parts)
Random variables X and Y have joint pdf fX,Y (x, y) that is constant in the shaded region (and zero elsewhere).
x
y
(a) Make a fully labeled sketch of the density fX (x). What is the mean and variance of X?
(b) Determine X̂ M M SE (y), the minimum mean square error estimator for X given the obser- vation Y = y. What is E[(X − X̂ M M SE (Y ))^2 ], the associated mean squared error?
(c) Determine X̂ LM M SE (y), the linear minimum mean square error estimator for X given the observation Y = y. What is E[(X − X̂ LM M SE (Y ))^2 ], the associated mean squared error?
(d) Consider a nonlinear estimator of the form X̂ (y) = αy^2 + βy + γ. Choose the constants α, β and γ such that the mean squared error E[(X − X̂ (Y ))^2 ] is minimized. Find the associated mean squared error.
Problem 4 (8/50, equally weighted parts)
Let Θ be uniformly distributed in the interval [0, 2 π] and consider the sequence of random variables Xn = cos(nΘ) for n = 1, 2 , ... The following two parts can be answered independently.
Part A. Does Xn converge almost surely (a.s.) as n tends to infinity? Justify your answer.
Part B. Does Xn converge in the mean square (m.s.) sense as n tends to infinity? Justify your answer. [Hint: Recall that sin a sin b = 12 [cos(a − b) − cos(a + b)] and cos a cos b = 1 2 [cos(a^ −^ b) + cos(a^ +^ b)].]
Bonus Problem (5/50)
Suppose that X 0 is a scalar random variable with zero mean and known variance cov(X 0 ) = E[X 02 ] = σ^2 X 0 and that W 0 , W 1 , W 2 , ... is a sequence of independent, identically distributed (i.i.d.) random variables with zero mean and known variance cov(Wi) = E[W (^) i^2 ] = σ W^2. Fur- thermore, assume that X 0 and the Wi’s are independent. Given that
Xk+1 = αXk + βWk
for k = 0, 1 , 2 , ..., find a recursive method for computing the variance of Xk. Also, derive conditions under which the mean and variance of Xk converge to steady state values. What are these steady state values?
