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

By Mark A. Pinsky, Samuel Karlin

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The system fails at the first component failure. (a) What is the mean time to failure for component A? For component B? (b) What is the mean time to system failure? (c) What is the probability that it is component A that causes system failure? (d) Suppose that it is component A that fails first. What is the mean remaining operating life of component B? 5 Consider a post office with two clerks. John, Paul, and Naomi enter simultaneously. John and Paul go directly to the clerks, while Naomi must wait until either John or Paul is finished before she begins service.

And having the probability mass function pN (n) = Pr{N = n} for n = 0, 1, . . Define the random sum X by X= 0 ξ1 + · · · + ξN if N = 0, if N > 0. 22) to simply X = ξ1 + · · · + ξN , understanding that X = 0 whenever N = 0.

Using the law of total probability, we obtain ∞ pX|N (k|n)pN (n) Pr{X = k} = n=0 ∞ = n=1 (n + k − 1)! n p (1 − p)k (1 − β)β n−1 k! (n − 1)! ∞ n+k−1 (βp)n−1 k = (1 − β)(1 − p)k p n=1 = (1 − β)(1 − p) p(1 − βp)−k−1 k = p − βp 1 − βp 1−p 1 − βp k for k = 0, 1, . . We recognize the marginal distribution of X as being of geometric form. Let g be a function for which the expectation of g(X) is finite. 4) 50 An Introduction to Stochastic Modeling and the conditional mean is not defined at values y for which pY (y) = 0.

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