1. Suppose events occur in time according to a Poisson Process with rate 1 per minute. (a) State the distribution of X, the number of events occuring in a one-hour time period, and hence show that the probability that no events occur in this one-hour time period is e-601. (b) The process starts at time 0. Let the time to the first event be Y minutes. State the distribution of Y and hence, or otherwise, find the probability that the first event occurs after 60 minutes. (c) The kth event (k 2 2) occurs at time Z minutes. Explain clearly how the following result can be used in the context of this Poisson Process to confirm that Z has a gamma distribution. State the parameters of this distribution. Result: if X, X2,...,X, are independent exponential random variables each with mean u, then E-1 Xi Gamma(n, u).

1. Suppose events occur in time according to a Poisson Process with rate 1 per minute. (a) State the distribution of X, the number of events occuring in a one-hour time period, and hence show that the probability that no events occur in this one-hour time period is e-601. (b) The process starts at time 0. Let the time to the first event be Y minutes. State the distribution of Y and hence, or otherwise, find the probability that the first event occurs after 60 minutes. (c) The kth event (k 2 2) occurs at time Z minutes. Explain clearly how the following result can be used in the context of this Poisson Process to confirm that Z has a gamma distribution. State the parameters of this distribution. Result: if X, X2,...,X, are independent exponential random variables each with mean u, then E-1 Xi Gamma(n, u).

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter14: Counting And Probability

Section14.2: Probability

Problem 3E: The conditional probability of E given that F occurs is P(EF)=___________. So in rolling a die the...

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