2.15. a) Prove that if YE(A), then for all t and s, P{Y>t+sY>t} = P{Y > s}. (2.16) b) Prove that if Y is absolutely continuous, positive and satisfies (2.16), then Y has an exponential distribution. [Hint: Prove that the survivor function Sy(t) = P{Y> t} fulfills the func- tional equation Sy(t + 8) = Sy(t) Sy(s) for s,t> 0.]
2.15. a) Prove that if YE(A), then for all t and s, P{Y>t+sY>t} = P{Y > s}. (2.16) b) Prove that if Y is absolutely continuous, positive and satisfies (2.16), then Y has an exponential distribution. [Hint: Prove that the survivor function Sy(t) = P{Y> t} fulfills the func- tional equation Sy(t + 8) = Sy(t) Sy(s) for s,t> 0.]
Chapter6: Exponential And Logarithmic Functions
Section6.8: Fitting Exponential Models To Data
Problem 56SE: Recall that the general form of a logistic equation for a population is given by P(t)=c1+aebt , such...
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![2.15. a) Prove that if Yª E(X), then for all t and s,
P{Y >t+s]Y > t} = P{Y > s}.
(2.16)
b) Prove that if Y is absolutely continuous, positive and satisfies
(2.16), then Y has an exponential distribution. [Hint: Prove
that the survivor function Sy(t) = P{Y > t} fulfills the func-
tional equation Sy(t+s) = Sy(t) Sy(s) for s, t > 0.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3dd9cd24-c83f-445a-b044-da2d154b184b%2F015105bb-70fe-4b5f-bd87-89180dce5706%2F8hpsk6l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2.15. a) Prove that if Yª E(X), then for all t and s,
P{Y >t+s]Y > t} = P{Y > s}.
(2.16)
b) Prove that if Y is absolutely continuous, positive and satisfies
(2.16), then Y has an exponential distribution. [Hint: Prove
that the survivor function Sy(t) = P{Y > t} fulfills the func-
tional equation Sy(t+s) = Sy(t) Sy(s) for s, t > 0.]
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Please solve letter b.
![2.15. a) Prove that if Yª E(X), then for all t and s,
P{Y >t+s]Y > t} = P{Y > s}.
(2.16)
b) Prove that if Y is absolutely continuous, positive and satisfies
(2.16), then Y has an exponential distribution. [Hint: Prove
that the survivor function Sy(t) = P{Y > t} fulfills the func-
tional equation Sy(t+s) = Sy(t) Sy(s) for s, t > 0.]](https://content.bartleby.com/qna-images/question/3dd9cd24-c83f-445a-b044-da2d154b184b/560adf28-d253-43b1-bc0c-384389c44e9e/nguqo5_processed.jpeg)
Transcribed Image Text:2.15. a) Prove that if Yª E(X), then for all t and s,
P{Y >t+s]Y > t} = P{Y > s}.
(2.16)
b) Prove that if Y is absolutely continuous, positive and satisfies
(2.16), then Y has an exponential distribution. [Hint: Prove
that the survivor function Sy(t) = P{Y > t} fulfills the func-
tional equation Sy(t+s) = Sy(t) Sy(s) for s, t > 0.]
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