Lecture 17 - More Distributions

Lecture 17 - More Distributions - 1 POISSON DISTRIBUTION...

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Unformatted text preview: 1 POISSON DISTRIBUTION Lecture 17 ORIE3500/5500 Summer2011 Li 1 Poisson distribution A random variable, taking on one of the values 0, 1, 2, ··· , is said to be a Poisson random variable with parameter λ , if its probability mass function is given by p X ( k ) = e- λ λ k k ! ,k = 0 , 1 , 2 ,.... The Poisson random variable , X ∼ Poi ( λ ), has mgf φ X ( t ) = ∞ X k =0 e tk e- λ λ k k ! = e- λ ∞ X k =0 ( e t λ ) k k ! = e- λ e e t λ = e λ ( e t- 1) . Let us compute the first and second moment φ X ( t ) = λe t e λ ( e t- 1) , φ 00 X ( t ) = λe t e λ ( e t- 1) + λ 2 e 2 t e λ ( e t- 1) . Hence E ( X ) = λ,E ( X 2 ) = λ + λ 2 ,var ( X ) = λ. 1 2 CHI-SQUARE DISTRIBUTION By arguments we have used several times earlier we can check that sum of independent Poisson random variables is also a Poisson random variable . If X 1 ,...,X n are independent and X i ∼ Poi ( λ i ), then φ X 1 + ··· + X n ( t ) = e ( λ 1 + ··· + λ n )( e t- 1) . Thus X 1 + ··· + X n ∼ Poi ( λ 1 + ··· + λ n ). Example Suppose that the average number of accidents occurring weekly on a particu- lar stretch of a highway follows a poisson distribution with λ = 3. Calculate the probability that there is at least one accident this week....
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Lecture 17 - More Distributions - 1 POISSON DISTRIBUTION...

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