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 light bulbs follow the exponential distribution with means 1000 hours, and 800 and 600 re- spectively. Assuming that the lifetimes are in- dependent, compute the expected time until one of the light bulbs burns out, the probability that the first one to burn out is the bulb with the longest expected lifetime.
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The Gamma random variable
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