(a) Prove that for any p = [0, 1] and any integers k and n such...

 

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(a) Prove that for any p = [0, 1] and any integers k and n such that 1 k n, n -p)--(-)- (-1)p(1 - p). n (m) p (1 - p)" m=k = m=k For this you may consider a binomial process with parameter p and its sequence of "arrival times" (the number of trials up to and including the kth success). (b) For any positive integer m, let fm be the density of the Erlang distribution of order m and parameter A and Fm be its distribution function. Express Fk in terms of f1,..., fk. (c) Let {Xt, t> 0} and {Yt, t> 0} be two independent Poisson processes with respective rates, A and . Let S denote the time of the kth arrival of the process X, and Tk that of the process Y. Obtain P(S; (a) Prove that for any p = [0, 1] and any integers k and n such that 1 k n, n -p)--(-)- (-1)p(1 - p). n (m) p (1 - p)" m=k = m=k For this you may consider a binomial process with parameter p and its sequence of "arrival times" (the number of trials up to and including the kth success). (b) For any positive integer m, let fm be the density of the Erlang distribution of order m and parameter A and Fm be its distribution function. Express Fk in terms of f1,..., fk. (c) Let {Xt, t> 0} and {Yt, t> 0} be two independent Poisson processes with respective rates, A and . Let S denote the time of the kth arrival of the process X, and Tk that of the process Y. Obtain P(S;

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a To prove the given equation we can utilize the concept of a binomial process with parameter p and ... View the full answer