09_22 - ( , = 1 / ) distribution, where is an integer, then...

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STAT 400 Examples for 09/22/2011 Fall 2011 Uniform Distribution over an interval [ a , b ] : For Uniform distribution, P ( c X d ) = d c b a - - , a c d b. E ( X ) = a b 2 + , Var ( X ) = 12 a b 2 ) ( - . Exponential Distribution : ( ) - = otherwise 0 0 for θ 1 θ x x f x e ( ) - = otherwise 0 0 for λ λ x x f x e E ( X ) = θ , Var ( X ) = θ 2 . E ( X ) = 1 / λ , Var ( X ) = 1 / λ 2 .
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1. Suppose the lifetime of a particular brand of light bulbs is exponentially distributed with mean of 400 hours. a) Find the probability that a randomly selected light bulb would last over 500 hours. b) Find the probability that 3 out of 7 randomly selected light bulbs would last over 500 hours. c) Find the probability that a randomly selected light bulb would last between 300 hours and 800 hours. Gamma Distribution : ( ) ( ) θ 1 α 1 α θ α x e x x f - - Γ = , 0 x < E ( X ) = α θ Var ( X ) = α θ 2 ( ) ( ) x e x x f λ 1 α α α λ - - Γ = , 0 x < E ( X ) = α / λ , Var ( X ) = α / λ 2 If X has a Gamma
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Unformatted text preview: ( , = 1 / ) distribution, where is an integer, then P ( X &gt; t ) = P ( Y t 1 ) , P ( X t ) = P ( Y t ) , where Y t has a Poisson ( t ) distribution. 2. Traffic accidents at a particular intersection follow Poisson distribution with an average rate of 1.4 per week. a) What is the probability that the next accident will not occur for three days? b) What is the probability that the next accident will occur during the third day? (That is, the time until the next accident is more than two days, but less than three days.) c) What is the probability that the second accident will occur before the end of the third day? d) What is the probability that the third accident will occur during the fourth day?...
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This note was uploaded on 11/07/2011 for the course STAT 400 taught by Professor Kim during the Fall '08 term at University of Illinois, Urbana Champaign.

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09_22 - ( , = 1 / ) distribution, where is an integer, then...

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