08 Poisson, Exponential, and Gamma distributions

08 Poisson, Exponential, and Gamma distributions -...

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Unformatted text preview: University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Poisson, Gamma, and Exponential distributions • A. Relation of Poisson and exponential distribution: Suppose that events occur in time according to a Poisson process with parameter λ . So X ∼ Poisson ( λ ). Let T denote the length of time until the first arrival. Then T is a continuous random variable. To find the probability density function (pdf) of T we begin with the cumulative distribution function (cdf) of T as follows: F ( t ) = P ( T ≤ t ) = 1- P ( T > t ) = 1- P ( X = 0) In words: The probability that we observe the first arrival after time t is the same as the probability that we observe no arrivals from now until time t . But X is Poisson with parameter λ which has parameter λt over the time interval (0 ,t ). We compute the above using: F ( t ) = 1- ( λt ) e- λt 0!...
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This note was uploaded on 01/14/2011 for the course STATS 100A taught by Professor Wu during the Fall '07 term at UCLA.

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08 Poisson, Exponential, and Gamma distributions -...

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