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Problem 5. Suppose the lifetime of a certain machine part is measured in months and has an exponential distribution with an expected lifetime of two months. Suppose the part is installed at time t = 0 and fails at time X, and is immediately replaced with a new part with a lifetime that is exponentially distributed with

Problem 5. Suppose the lifetime of a certain machine part is measured in months and has an exponential distribution with an expected lifetime of two months. Suppose the part is installed at time t = 0 and fails at time X, and is immediately replaced with a new part with a lifetime that is exponentially distributed with

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Problem 5. Suppose the lifetime of a certain machine part is measured in months and has an exponential distribution with an expected lifetime of two months. Suppose the part is installed at time t = 0 and fails at time X, and is immediately replaced with a new part with a lifetime that is exponentially distributed with parameter A2 = 1. The lifetime of the new part is independent of the lifetime of the old part. Let Z be the time at which the new part fails, counting from time t = 0. (a) Find the conditional probability density function of X given that Z = z. (b) Show that the conditional probability density function of X given that Z = z is a valid probability density function.
Transcribed Image Text:Problem 5. Suppose the lifetime of a certain machine part is measured in months and has an exponential distribution with an expected lifetime of two months. Suppose the part is installed at time t = 0 and fails at time X, and is immediately replaced with a new part with a lifetime that is exponentially distributed with parameter A2 = 1. The lifetime of the new part is independent of the lifetime of the old part. Let Z be the time at which the new part fails, counting from time t = 0. (a) Find the conditional probability density function of X given that Z = z. (b) Show that the conditional probability density function of X given that Z = z is a valid probability density function.
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