An Introduction to Stochastic Modeling, Fourth Edition by Mark A. Pinsky, Samuel Karlin

By Mark A. Pinsky, Samuel Karlin

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The mean lifetimes are E[X0 ] = 1000 and E[X1 ] = 1. Suppose we observe that the time of the first death is rather small, say, U = min{X0 , X1 } = 21 . In spite of vast disparity between the mean lifetimes, the observation that U = 12 provides no information about which of the two units, 0 or 1, was first to die! This apparent paradox is yet another, more subtle, manifestation of the memoryless property unique to the exponential density. We continue with the exercise. (d) Pr{W > t|N = 0} = e−λ1 t , t ≥ 0.

00002-2 c 2011 Elsevier Inc. All rights reserved. 48 An Introduction to Stochastic Modeling We are given the conditional probability mass function pX|N (k|n) = n k p (1 − p)n−k , k k = 0, 1, . . , n, and the marginal distribution pN (n) = M n q (1 − q)M−n , n n = 0, 1, . . , M. 3) to obtain M pX|N (k|n)pN (n) Pr{X = k} = n=0 M = n=k = M! n! pk (1 − p)n−k qn (1 − q)M−n k! (n − k)! n! (M − n)! q M! k p (1 − q)M k! 1−q × q 1−q k M n=k 1 (1 − p)n−k (n − k)! (M − n)! n−k = M! q(1 − p) ( pq)k (1 − q)M−k 1 + k!

7), we obtain Pr{W > t|N = 0} = Pr{W > t, N = 0} = e−λ1 t , Pr{N = 0} t ≥ 0, as desired. Of course a parallel formula holds conditional on N = 1: Pr{W > t|N = 1} = e−λ0 t , t ≥ 0, and using the law of total probability we obtain the distribution of W in the form Pr{W > t} = Pr{W > t, N = 0} + Pr{W > t, N = 1} λ1 λ0 e−λ1 t + e−λ0 t , t ≥ 0, = λ0 + λ1 λ0 + λ1 40 An Introduction to Stochastic Modeling (e) U and W = V − U are independent random variables. To establish this final consequence of the memoryless property, it suffices to show that Pr{U > u and W > w} = Pr{U > u} Pr{W > w} for all u ≥ 0, w ≥ 0.

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