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)
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Please do the following questions with handwritten working out.
![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)!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba18de34-fc06-47a6-b1ea-c54726b84874%2Fd0c34cbd-f883-44f0-86c7-1493527267ff%2Fmgmv4cd_processed.jpeg&w=3840&q=75)
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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