a. Explanation The failure rate (hazard rate) function is defined as follows: r ( t ) = f ( t ) 1 − F ( t ) The density function of an exponential distribution with β is as follows: f ( y ) = 1 β e − y β , y > 0 The distribution function for an exponential distribution can be obtained as follows: F ( t ) = P ( Y ≤ t ) = ∫ 0 t f ( y ) d y = ∫ 0 t 1 β e − y β d y = 1 β ∫ 0 t e − y β d y = 1 β [ − e − y β b. To determine Prove that for a Weibull distribution with m > 1 , r ( t ) is an increasing function of t .

Mathematical Statistics with Applications

7th Edition

ISBN: 9780495110811

Author: Dennis Wackerly, William Mendenhall, Richard L. Scheaffer

Publisher: Cengage Learning

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Chapter 4, Problem 190SE

To determine

Show that for an exponential density function r(t) is constant.

To determine

Prove that for a Weibull distribution with m>1, r(t) is an increasing function of t.

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Chapter 4, Problem 189SE

Chapter 4, Problem 191SE

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Show that for an exponential density function r ( t ) is constant.

Solution Summary: The author proves that for an exponential density function r(t) is constant. The failure rate (hazard rate) function is defined as follows.