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Assume the continuous RV X, with PDF fx and CDF Fx, measures the lifetime of some system or component. The hazard rate function for such a variable is defined as fx (t) h(t) : P(X > t) fx(t) 1 – Fx(t) and it measures the propensity of the system to fail shortly after time t, given that it has survived until time t. (a) Show that the hazard rate function for the Exponential(A) distribution for t > 0 is constant and equal to A (this is equivalent to the memoryless property). (b) Consider the Gamma(a = 2, X) distribution, which arises as the sum of two independent Exponential(A) RVs. Find the hazard rate function for this distribution for t > 0. (Hint: Use integration by parts for finding the CDF. )

Assume the continuous RV X, with PDF fx and CDF Fx, measures the lifetime of some system or component. The hazard rate function for such a variable is defined as fx (t) h(t) : P(X > t) fx(t) 1 – Fx(t) and it measures the propensity of the system to fail shortly after time t, given that it has survived until time t. (a) Show that the hazard rate function for the Exponential(A) distribution for t > 0 is constant and equal to A (this is equivalent to the memoryless property). (b) Consider the Gamma(a = 2, X) distribution, which arises as the sum of two independent Exponential(A) RVs. Find the hazard rate function for this distribution for t > 0. (Hint: Use integration by parts for finding the CDF. )

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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Assume the continuous RV X , with PDF fx and CDF Fx, measures the lifetime of some system or component. The hazard rate function for such a variable is defined as fx (t) P(X >t) fx(t) 1 – Fx(t) h(t) : and it measures the propensity of the system to fail shortly after time t, given that it has survived until time t. (a) Show that the hazard rate function for the Exponential(A) distribution for t > 0 is constant and equal to A (this is equivalent to the memoryless property). (D) Consider the Gamma(a = 2, X) distribution, which arises as the sum of two independent Exponential(A) RVs. Find the hazard rate function for this distribution fort>0. (Hint: Use integration by parts for finding the CDF. )
Transcribed Image Text:Assume the continuous RV X , with PDF fx and CDF Fx, measures the lifetime of some system or component. The hazard rate function for such a variable is defined as fx (t) P(X >t) fx(t) 1 – Fx(t) h(t) : and it measures the propensity of the system to fail shortly after time t, given that it has survived until time t. (a) Show that the hazard rate function for the Exponential(A) distribution for t > 0 is constant and equal to A (this is equivalent to the memoryless property). (D) Consider the Gamma(a = 2, X) distribution, which arises as the sum of two independent Exponential(A) RVs. Find the hazard rate function for this distribution fort>0. (Hint: Use integration by parts for finding the CDF. )
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