NotesOct8 - 2. Continuous models on unbounded intervals The...

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2. Continuous models on unbounded intervals The Exponential random variable with pa- rameter λ > 0 takes values in (0 , ) and has the density f X ( x ) = ( λe - λx if x 0 0 if x < 0. Notation : X Exp( λ ).
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Lack of memory of exponential random vari- able If X Exp( λ ), then for any x,y > 0 P ( X > x + y | X > x ) = P ( X > y ) = e - λy .
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Competition between exponential random variables Let X 1 ,X 2 ,...,X n be independent exponential random variables, X i Exp( λ i ) , i = 1 , 2 ,...,n . Let Y = min( X 1 ,X 2 ,...,X n ). Which one of the X i is the smallest (equal to Y )? What is the distribution of Y ?
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Distribution of Y : Y Exp( λ 1 + ... + λ n ) .
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The probability that X i is the smallest is proportional to its parameter λ i : P (min( X 1 ,X 2 ,...,X n ) = X i ) = λ i λ 1 + λ 2 + ... + λ n for i = 1 , 2 ,...,n , and P (min( X 1 ,X 2 ,...,X n ) = X i | Y = y ) = λ i λ 1 + λ 2 + ... + λ n for i = 1 , 2 ,...,n .
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Example : Suppose that the lifetimes of 3
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NotesOct8 - 2. Continuous models on unbounded intervals The...

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