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Hw7_ORIE361_2008 - ORIE 361/523 Homework 6 Instructor Mark...

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ORIE 361/523 – Homework 6 Instructor: Mark E. Lewis due March 13, 2008 (drop box) 1. A machine has two critically important parts and is subject to 3 different types of shocks. Shocks of type i occur at times of a Poisson process with rate λ i . Shocks of types 1 break part 1, those of type 2 break part 2 while those of type 3 break both parts. Let U and V be the failure times of the two parts. (a) Find P [ U > s, V > t ] . (b) Let t 0 and s 0 in the previous part to show that U and V each have exponential distributions. (c) Are U and V independent? 2. Let { N ( t ) , t 0 } be a Poisson process with rate λ , that is independent of the non-negative random variable T with mean μ and variance σ 2 . Find (a) Cov( T, N ( T )) (b) Var( N ( T )) 3. Consider n components with independent lifetimes which are such that component i functions for an exponential time with rate λ i . Suppose that all components are initially in use and remain so until they fail.
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