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part39 - EE 5375/7375 Random Processes October 7 2003...

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EE 5375/7375 Random Processes October 7, 2003 Homework #5 Solutions Problem 1. textbook problem 6.2 A discrete-time random process X n is defined as follows. A fair coin is tossed. If the outcome is heads, X n = 1 for all n ; if the outcome is tails, X n = - 1 for all n . (a) Sketch some sample paths of the process. (b) Find the PMF for X n . (c) Find the joint PMF for X n and X n + k . (d) Find the mean and autocovariance functions of X n . (a) If the outcome is heads, then X n = 1 , 1 , 1 , . . . . If the outcome is tails, then X n = - 1 , - 1 , - 1 , . . . (b) Since a head or tail is equally likely, it is equally likely that the process will be +1 or -1: P ( X n = 1) = P ( X n = - 1) = 1 / 2 (c) In either case of heads or tails, X n and X n + k will have the same value: P ( X n = 1 , X n + k = 1) = P ( X n = - 1 , X n + k = - 1) = 1 2 In neither case will X n and X n + k have different values: P ( X n = 1 , X n + k = - 1) = P ( X n = - 1 , X n + k = 1) = 0 (d) Since X n is 1 or -1 with equal likelihood, the mean is E ( X n ) = 1 · ( 1 2 ) - 1 · ( 1 2 ) = 0. The autocovariance function is C ( n, n + k ) = E ( X n X n + k ) - E ( X n ) E ( X n + k ) = E ( X n X n + k ) = 1 · P ( X n = 1 , X n + k = 1) + ( - 1)( - 1) · P ( X n = - 1 , X n + k = - 1) = 1 2 + 1 2 = 1 Problem 2. textbook problem 6.21 Let S n denote a binomial counting process. (a) Show that P ( S n = j, S m = i ) = P ( S n = j ) P ( S m = i ). (b) Find P ( S n = j | S m = i ) where n > m . (c) Show that P ( S n = j | S m = i, S l = k ) = P ( S n = j | S m = i ) where n > m > l . (a) We know that S n has independent increments that follow a binomial distribution. That is, for n > m , the increment S n - S m is the sum of n - m Bernoulli trials, which has the same distribution as S n - m . Hence, P ( S n = j, S m = i ) = P ( S n - S m = j - i, S m = i ) = P ( S n - m = j - i ) P ( S m = i ) = P ( S n = j ) P ( S m = i ) The last step follows because S n - m is the sum of n - m Bernoulli trials whereas S n is the sum of n Bernoulli trials. (b) Given S m = i , the event S n = j is the same as the increment S n - S m = j - i . This increment is the sum of n - m Bernoulli trials, so it is a binomial: P ( S n = j | S m = i ) = P ( S n - S m = j - i ) = n - m j - i p j - i (1 - p ) n - m - ( j - i ) 1
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(c) We have to use the property of independent increments. P ( S n = j | S m = i, S l = k ) = P ( S n = j, S m = i, S l = k ) P ( S m = i, S l = k ) = P ( S n - S m = j - i, S m - S l = i - k, S l = k ) P ( S m - S l = i - k, S l = k ) = P ( S n - S m = j - i ) P ( S m - S l = i - k ) P ( S l = k ) P ( S m - S l = i - k ) P ( S l = k ) = P ( S n - S m = j - i ) = P ( S n = j | S m = i ) Problem 3. textbook problem 6.23 Consider the following moving average processes: Y n = 1 2 ( X n + X n - 1 ) , X 0 = 0 Z n = 2 3 X n + 1 3 X n - 1 , X 0 = 0 (a) Flip a coin 10 times to obtain a realization of a Bernoulli random process X n . Find the resulting realizations of Y n and Z n . (b) Repeat part (a) with X n given by the random step process introduced in Example 6.12 (i.e., D n = 2 I n - 1 where I n is the Bernoulli random process. At any time, the process takes value +1 with probability p and value -1 with probability 1 - p ). (c) Find the mean, variance,and covariance of Y n and Z n if X n is a Bernoulli random process. Are the sample means of Y n and Z n in part (a) close to their respective means? (d) Repeat part (c) if X n is the random step process.
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