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Unformatted text preview: 90 Chapter 3 Summary Exponential distribution T is said to have an exponential distribution with rate , or T = exponential( ), if P ( T t ) = 1- e- t 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 ) = e- t 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( i ). min( T 1 ,...,T n ) = exponential( 1 + n ). P ( T i = min( T 1 ,...,T n )) = i / ( 1 + + n ). Theorem. Let 1 , 2 ,... be independent exponential( ). The sum T n = 1 + + n has a gamma( n, ) distribution. That is, the density function of T n is given by f T n ( t ) = e- t ( t ) n- 1 ( n- 1)! for t 0 and 0 otherwise. Poisson distribution X has a Poisson distribution with rate , or X = Poisson( ), for short, if P ( X = n ) = e- n n !...
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This note was uploaded on 09/29/2011 for the course MATH 4740 taught by Professor Durrett during the Spring '10 term at Cornell University (Engineering School).
- Spring '10