Concise - A Concise Summary Everything you need to know...

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A Concise Summary Everything you need to know about exponential and Poisson Exponential Distribution Assume that X exp( λ ), by which we mean that X has an exponential distribution with rate λ . Then X has mean 1 ; i.e., EX = 1 . Also the variance is V ar ( X ) = ( EX ) 2 = 1 2 . In addition, assume that Y exp( μ ) and X i exp( λ i ) for i = 1 , ··· ,n , where all these exponential random variables are independent. 1. Lack of memory : P ( X > s + t | X > s ) = P ( X > t ) for all s > 0 and t > 0. (check the computation) 2. Minimum : min { X,Y } ∼ exp( λ + μ ) (check the computation) and hence min { X 1 , ··· ,X n } ∼ exp( λ 1 + ··· + λ n ) without computation. 3. Maximum : X + Y = min { X,Y } +max { X,Y } tells us an easy way to compute E [max { X,Y } ].) 4. More on Minimum : P ( X = min { X,Y } ) = P ( X < Y ) = λ λ + μ : (check the computation) and hence P ( X k = min { X 1 , ··· ,X n } ) = λ k λ 1 + ··· + λ n without computation. 5.
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