Northeastern University December, 2018

Department of Electrical and Computer Engineering EECE 7204 (Fall 2018)

Final Exam

This is a take-home exam that has to be submitted over Blackboard after completion. The due date is December 10th, 2018, at 22h.

Problem 1 (10/100): We have two coins: 1 and 2. While 1 is a fair coin, the second coin has  two  heads.  In  an  experiment,  we  select    1  with  probability  3/4  and    2  with  probability  1/4. The selected coin is tossed n times and n heads are observed.

(a) Compute the probability of having selected C2 given the outcome of the experiment.

(b) What is the limit of that probability when n → ∞ ?

Problem 2 (10/100): Let (X, Y ) be a 2-dimensional random variable with independent mar- ginal components X and Y which are exponential with expected value mX = mY = 1. Find the probabilities P (X Y ≥ 1) and P (X > 2Y |X > Y ).

Problem 3 (10/100): Let X and Y be two uncorrelated random variables and consider their sum S = X + Y and difference R = X Y . Find a necessary and sufficient condition for S and R to be also uncorrelated.

Problem 4 (10/100): Let X be a continuous random variable uniformly distributed in (0, 1).

Find the probability density functions of the random variables Y = g(X) when:

(a) g(x) = 8x3

(b) g(x) = (x − 1/2)2

Problem 5 (10/100): Two friends agree in meeting for a coffee between 9 and 10 in the morning. The times of arrivals at the cafeteria for each one are random and independent, being uniform in that interval. If we denote by X and Y the time of arrival of each friend, then T = X  Y is the random variable representing the time that the first person has to wait before the other friend arrives. Find E(T ), the expected waiting time, using these two methods:

(a) Compute the cumulative density function of T ; then obtain the probability density function of T ; and evaluate the expectation.

(b) Applying the expectation theorem.

Problem 6 (10/100): The random variables X and Y have a joint probability density function given by

fXY (x, y) = . 8xy 0 ≤ y x ≤ 1

(a) Obtain the conditional density of Y given X, that is fY |X (y, x).

(b) Compute the conditional expectation E(Y 4 X = x). Discuss why this quantity is a random variable.

Problem 7 (10/100): Obtain the moment generating function of an exponential random variable X with parameter µ. Using that result, show that E(Xn) = n!

Hint: 1 1 z = 1 + z + z2 + · · · + zn + · · · for −1 < z < 1

Problem 8 (10/100): Let Sn = min(X1, X2, . . . , Xn), where Xk k 1 is a sequence of uniform random variables that are independent and identically distributed in the interval [0, 1].

n−1

(a) Show that the probability density functions of Sn is fSn (s) = n(1 − s) in the interval

0 < s < 1 and zero otherwise. To do that, first obtain FSn (s), its probability distribution function.

(b) Prove that the sequence {Xn}n≥1 converges to 0 in mean square and in probability.

Problem 9 (10/100): A and B are two zero-mean, independent random variables with vari-

ances σ2 = σ2 = 2. Given those random variables, a stochastic process X(t) is defined as

X(t) = At + B .

(a) Find the mean and autocorrelation functions of the process X(t). Is it a strict-sense stationary process? Is it a wide-sense stationary process? Justify your answer.

Assume that A and B are continuous random variables. Show that the first-order probability density function of the process, fX(x; t), can be expressed in terms of the probability density functions of the random variables as

where A fA(a) and B fB(b).

Problem 10 (10/100): Let A and B two uncorrelated, zero-mean random variables with identical variance σ2. A stochastic process is constructed such that X(t) = A cos(ω0t) + B cos(ω0t), where ω0 is a known constant. Is X(t) wide-sense stationary? Is X(t) ergodic in the mean?