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7.1.1 Verify the following equivalences for the age and the excess life in a renewal process N(t): (Assume t> x.) Pr{8x, y} = Pr{N(t-x) = N(t+y)} = Pr{Wkt+y} k=0 = [1 - F(t+y)] t-x ΣΤ +Σ [1 − F(t+y−z)]dFk(z). k=10 Carry out the evaluation when the interoccurrence times are exponentially dis- tributed with parameter λ, so that dFk is the gamma density xkzk-1 e dFk (z) = -zdz for z> 0. (k-1)!

7.1.1 Verify the following equivalences for the age and the excess life in a renewal process N(t): (Assume t> x.) Pr{8x, y} = Pr{N(t-x) = N(t+y)} = Pr{Wkt+y} k=0 = [1 - F(t+y)] t-x ΣΤ +Σ [1 − F(t+y−z)]dFk(z). k=10 Carry out the evaluation when the interoccurrence times are exponentially dis- tributed with parameter λ, so that dFk is the gamma density xkzk-1 e dFk (z) = -zdz for z> 0. (k-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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7.1.1 Verify the following equivalences for the age and the excess life in a renewal process N(t): (Assume t> x.) Pr{8x, y} = Pr{N(t-x) = N(t+y)} = Pr{Wk<t-x, Wk+1 >t+y} k=0 = [1 - F(t+y)] t-x ΣΤ +Σ [1 − F(t+y−z)]dFk(z). k=10 Carry out the evaluation when the interoccurrence times are exponentially dis- tributed with parameter λ, so that dFk is the gamma density xkzk-1 e dFk (z) = -zdz for z> 0. (k-1)!
Transcribed Image Text:7.1.1 Verify the following equivalences for the age and the excess life in a renewal process N(t): (Assume t> x.) Pr{8x, y} = Pr{N(t-x) = N(t+y)} = Pr{Wk<t-x, Wk+1 >t+y} k=0 = [1 - F(t+y)] t-x ΣΤ +Σ [1 − F(t+y−z)]dFk(z). k=10 Carry out the evaluation when the interoccurrence times are exponentially dis- tributed with parameter λ, so that dFk is the gamma density xkzk-1 e dFk (z) = -zdz for z> 0. (k-1)!
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