40. Let X1, X2, . . . , Xn denote the first n interarrival times of a...
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40. Let X1, X2, . . . , Xn denote the first n interarrival times of a Poisson process and set Sn = ni
=1 Xi .
(a) What is the interpretation of Sn?
(b) Argue that the two events {Sn ≤ t } and {N(t ) ≥ n} are identical.
(c) Use part
(b) to show that
d) By differentiating the distribution function of Sn given in part (c), conclude that Sn is a gamma random variable with parameters n and λ. (This result also follows from Corollary 5.7.2.)
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Introduction To Probability And Statistics For Engineers And Scientists
ISBN: 9780125980579
3rd Edition
Authors: Sheldon M. Ross
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