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Poisson Processes 247 5.3.7 A critical component on a submarine has an operating lifetime that is expo- nentially distributed with mean 0.50 years. As soon as a component fails, it is replaced by a new one having statistically identical properties. What is the smallest number of spare components that the submarine should stock if it is leaving for a one-year tour and wishes the probability of having an inoperable unit caused by failures exceeding the spare inventory to be less than 0.02? 5.3.8 Consider a Poisson process with parameter . Given that X(t) = n events occur in time t, find the density function for W r , the time of occurrence of the rth event. Assume that r n. 5.3.9 The following calculations arise in certain highly simplified models of learn- ing processes. Let X 1 (t) and X 2 (t) be independent Poisson processes having parameters 1 and 2 , respectively. (a) What is the probability that X 1 (t) = 1 before X 2 (t) = 1? (b) What is the probability that X 1 (t) = 2 before X 2 (t) = 2? 5.3.10 Let {W n } be the sequence of waiting times in a Poisson process of intensity = 1. Show that X n = 2 n exp {-W n } defines a nonnegative martingale. 5.4 The Uniform Distribution and Poisson Processes The major result of this section, Theorem 5.7, provides an important tool for comput-