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ch3summary - 90 Chapter 3 Summary Exponential distribution...

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90 Chapter 3 Summary Exponential distribution T is said to have an exponential distribution with rate λ , or T = exponential( λ ), if P ( T t ) = 1 - e - λt for t 0. This distribution has the lack of memory property P ( T > t + s | T > t ) = P ( T > s ). The density function f T ( t ) = λe - λt for t 0 and 0 otherwise. The mean ET = 1 while the variance var ( T ) = 1 2 . Let T 1 , . . . , T n be independent with where T i = exponential( λ i ). min( T 1 , . . . , T n ) = exponential( λ 1 + · · · λ n ). P ( T i = min( T 1 , . . . , T n )) = λ i / ( λ 1 + · · · + λ n ). Theorem. Let τ 1 , τ 2 , . . . be independent exponential( λ ). The sum T n = τ 1 + · · · + τ n has a gamma( n, λ ) distribution. That is, the density function of T n is given by f T n ( t ) = λe - λt · ( λt ) n - 1 ( n - 1)! for t 0 and 0 otherwise. Poisson distribution X has a Poisson distribution with rate λ , or X = Poisson( λ ), for short, if P ( X = n ) = e - λ λ n n ! for n = 0 , 1 , 2 , . . . Theorem. For any k 1, EX ( X - 1) · · · ( X - k + 1) = λ k . Hence EX = λ and var ( X ) = λ . Theorem. If X i are independent Poissson( λ i ) then X 1 + · · · + X k = Poisson( λ 1 + · · · + λ n ) .
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