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NotesOct8

# 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 .
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 ) .
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.
The Gamma random variable

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• '10
• SAMORODNITSKY
• Probability theory, Exponential distribution, Erlang distribution

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