We have seen that if E(X1) = E(X2) = … =E(Xn) = µ, then E(X1 +…+Xn) = nµ. In some applications, the number of Xi’s under consideration is not a fixed number n but instead is an rv N. For example, let N = the number of components that are brought into a repair shop on a particular day, and let Xi denote the repair shop time for the ith component. Then the total repair time is X1 + X2 + … XN, the sum of a random number of random variables. When N is independent of the Xi’s, it can be shown that
E(X1 + … + XN) = E(N) · µ
a. If the expected number of components brought in on a particularly day is 10 and expected repair time for a randomly submitted component is 40 min, what is the expected total repair time for components submitted on any particular day?
b. Suppose components of a certain type come in for repair according to a Poisson process with a rate of 5 per hour. The expected number of defects per component is 3.5. What is the
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Probability and Statistics for Engineering and the Sciences
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage