# Stochastic

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Unformatted text preview: stock at time t and suppose that at times of a Poisson process with rate the price is multiplied by a random variable Xi > 0 with mean µ and variance 2 . That is, N (t) St = S0 Y Xi i=1 where the product is 1 if N (t) = 0. Find ES (t) and var S (t). 2.39. Messages arrive to be transmitted across the internet at times of a Poisson process with rate . Let Yi be the size of the ith message, measured in bytes, and let g (z ) = Ez Yi be the generating function of Yi . Let N (t) be the number of arrivals at time t and S = Y1 + · + YN (t) be the total size of the messages up to time t. (a) Find the generating function f (z ) = E (z S ). (b) Di↵erentiate and set z = 1 to ﬁnd ES . (c) Di↵erentiate again and set z = 1 to ﬁnd E {S (S 1)}. (d) Compute var (S ). 2.40. Let {N (t), t 0} be a Poisson process with rate . Let T independent with mean µ and variance 2 . Find cov (T, NT ). 0 be an 2.41. Let t1 , t2 , . . . be independent exponential( ) random variables and let N be an independent random variable with P (N = n) = (1 p)n 1 . What is the distribution of the random sum T = t1 + · · ·...
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## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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