soln9 - Probability and Stochastic Processes: A Friendly...

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Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J. Goodman Problem Solutions : Yates and Goodman,7.1.2 7.4.1 7.4.2 7.5.1 7.6.1 8.1.1 6.5.1 6.5.2 and 6.5.3 Problem 7.1.2 (a) Since Y = X 1 +( - X 2 ) , Theorem 7.1 says that the expected value of the difference is E [ Y ] = E [ X 1 ]+ E [ - X 2 ] = E [ X ] - E [ X ] = 0 (b) By Theorem 7.2, the variance of the difference is Var [ Y ] = Var [ X 1 ]+ Var [ - X 2 ] = 2Var [ X ] Problem 7.4.1 K i has PMF P K ( k ) = 1 - p k = 0 p k = 1 0 otherwise (a) The MGF of K is φ K ( s ) = E £ e sK ¤ = 1 - p + pe s (b) By Theorem 7.10, M = K 1 + K 2 + ... + K n has MGF φ M ( s ) = [ φ K ( s )] n = [ 1 - p + pe s ] n (c) Although we could just use the fact that the expectation of the sum equals the sum of the ex- pectations, the problem asks us to find the moments using φ M ( s ) . In this case, E [ M ] = d φ M ( s ) ds ¯ ¯ ¯ ¯ s = 0 = n ( 1 - p + pe s ) n - 1 pe s ¯ ¯ s = 0 = np (d) The second moment of M can be found via E £ M 2 ¤ = d φ M ( s ) ds ¯ ¯ ¯ ¯ s = 0 = np ( ( n - 1 )( 1 - p + pe s ) pe 2 s +( 1 - p + pe s ) n - 1 e s ¯ s = 0 = np [( n - 1 ) p + 1 ] The variance of M is Var [ M ] = E £ M 2 ¤ - ( E [ M ]) 2 = np ( 1 - p ) = n Var [ K ] 1
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Problem 7.4.2 points
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soln9 - Probability and Stochastic Processes: A Friendly...

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