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4.97 Suppose that the positive random variables X₁,..., X, are independent of each other and have respective failure-rate functions r₁(x),...,n(x). Prove that the random variable V = min(X₁,..., Xn) has the failure-rate function k-1 rk (x). In particular, conclude that V has an exponential distribution with parameter k-1 Ak when each of the X; is exponen- tially distributed with parameter ₁.

4.97 Suppose that the positive random variables X₁,..., X, are independent of each other and have respective failure-rate functions r₁(x),...,n(x). Prove that the random variable V = min(X₁,..., Xn) has the failure-rate function k-1 rk (x). In particular, conclude that V has an exponential distribution with parameter k-1 Ak when each of the X; is exponen- tially distributed with parameter ₁.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Question number 4.97

ery on al ne 0, 01 4.11 Probability Distributions and Entropy 205 4.97 Suppose that the positive random variables X₁,..., Xn are independent of each other and have respective failure-rate functions r₁(x),...,n(x). Prove that the random variable V = min(X₁,..., Xn) has the failure-rate function 1 rk(x). In particular, conclude that V has an exponential distribution with parameter 1 Ak when each of the X; is exponen- tially distributed with parameter λi. = 4.98 Suppose that the positive random variable X has the failure-rate function (1 + x)−¹ for x > 0. Verify the remarkable result that the reliability function of X is also given by (1+x)-¹ for x > 0. 4.99 The lifetime X of a semiconductor has the failure-rate function r(x) = λax-¹/(1+x) for x ≥ 0, where the parameters a and λ are positive. 2016 THE What is the reliabilitu of the lifetime X? Verify that the failure-
Transcribed Image Text:ery on al ne 0, 01 4.11 Probability Distributions and Entropy 205 4.97 Suppose that the positive random variables X₁,..., Xn are independent of each other and have respective failure-rate functions r₁(x),...,n(x). Prove that the random variable V = min(X₁,..., Xn) has the failure-rate function 1 rk(x). In particular, conclude that V has an exponential distribution with parameter 1 Ak when each of the X; is exponen- tially distributed with parameter λi. = 4.98 Suppose that the positive random variable X has the failure-rate function (1 + x)−¹ for x > 0. Verify the remarkable result that the reliability function of X is also given by (1+x)-¹ for x > 0. 4.99 The lifetime X of a semiconductor has the failure-rate function r(x) = λax-¹/(1+x) for x ≥ 0, where the parameters a and λ are positive. 2016 THE What is the reliabilitu of the lifetime X? Verify that the failure-
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