Stochastic

Suppose that the machines fail at rate i what is the

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Unformatted text preview: said to have an exponential distribution with rate , or T = exponential( ), if P (T t) = 1 e t for all t 0. The mean is 1/ , variance 1/ 2 . The density function is fT (t) = e t . The sum of n independent exponentials has the gamma(n, ) density e t ( t)n 1 (n 1)! Lack of memory property. “if we’ve been waiting for t units of time then the probability we must wait s more units of time is the same as if we haven’t waited at all.” P (T > t + s|T > t) = P (T > s) Exponential races. Let T1 , . . . , Tn are independent, Ti = exponential( i ), and S = min(T1 , . . . , Tn ). Then S = exponential( 1 + · · · + n ) P (Ti = min(T1 , . . . , Tn )) = i 1 + ··· + n max{S, T } = S + T min{S, T } so taking expected value if S = exponential(µ) and T = exponential( ) then 1 1 1 + µ µ+ 1 1 = + ·+ µ+ +µ µ E max{S, T } = Poisson(µ) distribution. P (X = n) = e X are µ. µ 1 · +µ µ /n!. The mean and variance of µn Poisson process. Let t1 , t2 , . . . be independent exponential( ) random variables. Let Tn = t1 + . . . + tn be the time of the nth arrival. Let N (t) = max{n : T...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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