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ECE Ph.D. Qualifying Exam

Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

January 2004



Question

1. (30 pts.)

This question consists of two separate short questions relating to the structure of probability space:

(a)

Assume that $ \mathcal{S} $ is the sample space of a random experiment and that $ \mathcal{F}_{1} $ and $ \mathcal{F}_{2} $ are $ \sigma $ -fields (valid event spaces) on $ \mathcal{S} $ . Prove that $ \mathcal{F}_{1}\cap\mathcal{F}_{2} $ is also a $ \sigma $ -field on $ S $ .

(b)

Consider a sample space $ \mathcal{S} $ and corresponding event space $ \mathcal{F} $ . Suppose that $ P_{1} $ and $ P_{2} $ are both balid probability measures defined on $ \mathcal{F} $ . Prove that $ P $ defined by $ P\left(A\right)=\alpha_{1}P_{1}\left(A\right)+\alpha_{2}P_{2}\left(A\right),\qquad\forall A\in\mathcal{F} $ is also a valid probability measure on $ \mathcal{F} $ if $ \alpha_{1},\;\alpha_{2}\geq0 $ and $ \alpha_{1}+\alpha_{2}=1 $ .


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2. (10 pts.)

Identical twins come from the same egg and and hence are of the same sex. Fraternal twins have a probability $ 1/2 $ of being of the same sex. Among twins, the probability of a fraternal set is p and of an identical set is $ q=1-p $ . Given that a set of twins selected at random are of the same sex, what is the probability they are fraternal? (Simplify your answer as much as possible.) Sketch a plot of the conditional probability that the twins are fraternal given that they are of the same sex as a function of $ q $ (the probability that a set of twins are identical.)

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3. (30 pts.)

Let $ \mathbf{X}\left(t\right) $ be a real continuous-time Gaussian random process. Show that its probabilistic behavior is completely characterized by its mean $ \mu_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right] $ and its autocorrelation function $ R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}\left(t_{2}\right)\right]. $

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4. (30 pts.)

Assume that $ \mathbf{X}\left(t\right) $ is a zero-mean, continuous-time, Gaussian white noise process with autocorrelation function $ R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=\delta\left(t_{1}-t_{2}\right) $. Let $ \mathbf{Y}\left(t\right) $ be a new random process defined by $ \mathbf{Y}\left(t\right)=\frac{1}{T}\int_{t-T}^{t}\mathbf{X}\left(s\right)ds $, where $ T>0 $ .

(a)

What is the mean of $ \mathbf{Y}\left(t\right) $ ?

(b)

What is the autocorrelation function of $ \mathbf{Y}\left(t\right) $ ?

(c)

Write an expression for the second-order pdf $ f_{\mathbf{Y}\left(t_{1}\right)\mathbf{Y}\left(t_{2}\right)}\left(y_{1},y_{2}\right) $ of $ \mathbf{Y}\left(t\right) $ .

(d)

Under what conditions on $ t_{1} $ and $ t_{2} $ will $ \mathbf{Y}\left(t_{1}\right) $ and $ \mathbf{Y}\left(t_{2}\right) $ be statistically independent?

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