ps2.pdf - EE5137 Stochastic Processes Problem Set 2...

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EE5137 Stochastic Processes: Problem Set 2 Assigned: 24/08/17, Due: 31/08/17 1. Exercise 1.12 (Gallager’s book) Hint: For parts (a) express the CDF of M + (the maximum of the N rvs) in terms of the CDFs of the individual rvs. Part (b) is analogous. Part (c) is most challenging. You may first condition on the event { X 1 = x } . Then note that X 1 = M + iff X j x for all 2 j n . Also given X 1 = M + = x , we have R = M + - M - r iff X j > x - r for 2 j n . Now since the rvs are i.i.d., Pr( M + = X 1 , R r | X 1 = x ) = n Y j =2 Pr( x - r < X j x ) Continue the above argument (average over X 1 = x ) to show that Pr( R r ) = Z -∞ nf X ( x )[ F X ( x ) - F X ( x - r )] n - 1 d x. 2. Exercise 1.14 (Gallager’s book) 3. Exercise 1.20 (Gallager’s book) 4. (Optional) Exercise 1.16 (Gallager’s book)
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