COMER December 13, 2006

ECE 600 Final Exam

1. Enter your name and signature in the space provided on this page now. 2. You may not use a calculator or any reference materials. 3. Show all of your work. Partial credit will be given, at the discretion of the instructor.

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1. (25 points) There are two types of batteries in a bin. When in use, type 1 batteries last (in hours) an exponentially distributed time with rate 1 and type 2 batteries last an exponentially distributed time with rate 2. A battery that is randomly chosen from the bin will be a type i battery with probability pi, where p1+p2=1. If a randomly chosen battery is still operating after t hours of use, what is the probability that it will still be operating after an additional s hours?

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2. (30 points) Suppose that Xi, i=1,2,3, are independent Poisson random variables with respective means i, i=1,2,3. Let X = X1 + X2 and Y = X2 + X3.

(a) (10 points) Find E[X] and E[Y]. (b) (10 points) Find Cov(X,Y). (c) (10 points) Find the joint pmf P(X=i,Y=j).

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3. (30 points) Customers arrive at a certain retail establishment according to a Poisson process with rate per hour. Suppose that two customers arrive during the first hour. Find the probability that

(a) (15 points) both arrived in the first 20 minutes; (b) (15 points) at least one arrived in the first 30 minutes.

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4. (25 points) Let Y(t) be the output of a linear time-invariant system with impulse response h(t) and input X(t) + N(t), where X(t) and N(t) are jointly WSS independent random processes and N(t) has zero mean. Let Z(t) = X(t) Y(t). Find Sz() in terms of SX(), SN(), and H().

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5. (25 points) A moving average process is produced as follows:

1 1X W W Wn n n p n p = + + +

where Wn is a zero mean white noise process with RW(k) = E[WnWn+k] = 2(k), and 1,, p are real-valued constants. Find RX(k) = E[XnXn+k].