ECE Ph.D. Qualifying Exam

Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2016



Question

Part 1.

A friend tossed two fair coins, You asked "Did a coin land heads?" Your friends answers "yes." What is the probability that both coins landed heads? Justify your answer.

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Part 2.

A point $ \omega $ is picked at random in the triangle shown below (all points are equally likely.) let the random variable $ X(\omega) $ be the perpendicular distance from $ \omega $ to be base as shown in the diagram.
Wan82_CS1_problem.PNG
(a) Find the cumulative distribution function (cdf) of $ \mathbf{X} $.
(b) Find the probability distribution function (pdf) of $ \mathbf{X} $.
(c) Find the mean of $ \mathbf{X} $.
(d) What is the probability that $ \mathbf{X}>h/3 $.

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Part 3.

Let $ X $ and $ Y $ be independent, jointly-distributed Poisson random variables with means with mean $ \lambda $ and $ \mu $. Let $ Z $ be a new random variable defined as
$ Z=X+Y $
(a) Find the probability mass function (pmf) of $ \mathbf{Z} $.
(b) Show that the conditional probability mass function (pmf) of $ X $ conditioned on the event $ {Z=n} $ is binomially distributed, and determine the parameters of the binomial distribution (recall that there are two parameters $ "n" $ and $ "p" $) required to specify a binomial distribution $ b(n,p) $).

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Part 4.

Let $ X(t) $ be a wide-sense stationary Gaussian random process with mean $ \mu_x $ and autocorrelation function $ R_xx(\tau) $. Let
$ Y(t)=c_1X(t)-c_2X(t-T) $,
where $ c_1,c_2 $ and $ T $ are real numbers. What is the probability that $ Y(t) $ is less than or equal to a real number $ /\gamma? $ Express your answer in terms of $ c_1,c_2,\mu_x,\sigma_x^2 $, and $ R_xx(\tau), \gamma $ and the "phi function"
$ \Phi(x)=\int_{-\infty}^{x} \dfrac{1}{\sqrt{2\pi}}e^{-z^{2}/2} dz $

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