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# PS RAO 15 - Lecture 15 Exponential Distribution In Gamma...

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Lecture # 15 Exponential Distribution In Gamma Dist. Put α =1, we get f(x)= exp(-x / β )/ β ; x > 0, β > 0 = 0 ; e.w . f(x) α =1, β =1

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The distribution arises in practice in conjunction with the study of Poisson processes, where we have discrete events are being observed in continuous time interval. If we let W denote the time of the occurrence of the first event, then W is a continuous random variable Theorem : Consider a Poisson process with parameter λ . Let W denote the time of the occurrence of the first event. W has an Exponential distribution with β =1/ λ
Proof: This theorem is distribution of waiting time The distribution function F for W is given by ] w W [ P 1 ] w W [ P ) w ( F - = = Here, we note that, the first occurrence of the event will take place after time w only if no occurrence of the event are recorded in the time interval [0,w] . Let X denote the number of occurrences of the event in this time interval.

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X is a poisson random variable with parameter λ w. w 0 w e ! 0 ) w ( e ] 0 X [ P ] w W [ P , Thus λ - λ - = λ = = = By substitution, we get w e 1 ] w W [ P 1 ) w ( F λ - - = - = w e 1 ] w W [ P 1 ) w ( F λ - - = - =
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PS RAO 15 - Lecture 15 Exponential Distribution In Gamma...

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