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lec14 - Gamma Example(Baron 4.7 Compilation of a computer...

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Gamma Example (Baron 4.7) Compilation of a computer program consists of 3 blocks that are processed sequentially, one after the other. Each block takes Exponential time with mean of 5 minutes, independently of other blocks. (a) Compute the expectation and variance of the total compilation time. For a Gamma random variable T with α = 3 and λ = 1 / 5 , E ( T ) = 3 1 / 5 = 15 (min) and V ar ( T ) = 3 (1 / 5) 2 = 75 (min) (b) Compute the probability for the entire program to be compiled in less than 12 minutes. This can be done using repeated integration by parts (see Baron p.95). 1

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Gamma Example (Cont’d) However we will use the Gamma-Poisson formula: For T Gam ( α, λ ) and X Po ( λt ) , P ( T > t ) = P ( X < α ) and P ( T t ) = P ( X α ) Need P ( T < 12) where T Gam (3 , 1 / 5) . Note that t = 12 so X Po (12 / 5) i.e., X Po (2 . 4) . From the Gamma-Poisson formula, P ( T < 12) = P ( T 12) = P ( X 3) = 1 - Po 2 . 4 (2) = 1 - 0 . 5697 2
Erlang distribution Hits on a web page Recall: we modeled waiting times until the first hit as Exp (2) . How long do we have to wait for the second hit? To calculate waiting time for the second hit, we add the waiting times until the first hit and the time between the first and the second hit. Let Y 1 =the waiting time until the first hit. Then Y 1 Exponential with λ = 2 Let Y 2 , the time between first and second hit. By the memoryless property of the exponential distribution, Y 2 has the same distribution as waiting time for the first hit. That is Y 2 Exponential with λ = 2 We want the total time for the second hit X := Y 1 + Y 2 .

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