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Unformatted text preview: Stochastic Signals and Systems Homework 2 Book Solutions Problem Solutions : Yates and Goodman,2.3.2 2.3.9 2.8.10 2.9.6 4.2.1 4.3.3 4.4.2 4.4.11 4.7.5 and 4.7.1 Problem 2.3.2 (a) Each paging attempt is an independent Bernoulli trial with success probability p . The number of times K that the pager receives a message is the number of successes in n Bernoulli trials and has the binomial PMF P K ( k ) = ½ ( n k ) p k ( 1- p ) n- k k = , 1 ,... , n otherwise (b) Let R denote the event that the paging message was received at least once. The event R has probability P [ R ] = P [ B > ] = 1- P [ B = ] = 1- ( 1- p ) n To ensure that P [ R ] ≥ . 95 requires that n ≥ ln ( . 05 ) / ln ( 1- p ) . For p = . 8, we must have n ≥ 1 . 86. Thus, n = 2 pages would be necessary. Problem 2.3.9 The packets are delay sensitive and can only be retransmitted d times. For t < d , a packet is transmit- ted t times if the first t- 1 attempts fail followed by a successful transmission on attempt t . Further, the packet is transmitted d times if there are failures on the first d- 1 transmissions, no matter what the outcome of attempt d . So the random variable T , the number of times that a packet is transmitted, can be represented by the following PMF. P T ( t ) = p ( 1- p ) t- 1 t = 1 , 2 ,... , d- 1 ( 1- p ) d- 1 t = d otherwise Problem 2.8.10 We wish to minimize the function e ( ˆ x ) = E £ ( X- ˆ x ) 2 ¤ with respect to ˆ x . We can expand the square and take the expectation while treating ˆ x as a constant....
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- Spring '08
- Probability theory, dx