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lect11

# lect11 - MOMENTS AND FACTORIAL MOMENTS THE POISSON...

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MOMENTS AND FACTORIAL MOMENTS C Outline MOMENTS AND FACTORIAL MOMENTS THE POISSON DISTRIBUTION THE POISSON APPROXIMATION TO THE BINOMIAL CIRCUMSTANCES WHERE POISSON DISTRIBUTION APPLIES 1 / 13 Xinghua Zheng Lect 11: The Poisson distribution

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MOMENTS AND FACTORIAL MOMENTS C MOMENTS, FACTORIAL MOMENTS AND EXPONENTIAL FUNCTION For a random variable X , μ k := E ( X k ) is called the k th moment of X , ν k := E ( X ) k = E ( X ( X - 1 ) ... ( X - k + 1 )) is the called the k th factorial moment of X . The mean and variance of X can be written as E ( X ) = μ 1 = ν 1 . Var ( X ) = = . The exponential function e x = exp ( x ) can be deﬁned as e x = X k = 0 x k k ! = 1 + x 1 + x 2 2 ! + x 3 3 ! + ..., for any x ( -∞ , ) . satisﬁes e x · e y = e x + y , for any x , y ( -∞ , ) . 2 / 13 Xinghua Zheng Lect 11: The Poisson distribution
MOMENTS AND FACTORIAL MOMENTS C EXAMPLE: COUNTING DEATHS What is a reasonable chance model for the number of deaths in a large city on an ordinary day? The number n of people in the city is large. For now, suppose that each person has the same small chance p to die on that day. People die of one another. Let X be the number of deaths that day. Under the assumptions, X . The expected number of deaths is μ = E ( X ) = . Assume that this is of moderate size, so one is interested in P [ X = k ] for moderate values of k : P [ X = k ] = ± n k ² p k q n - k = 1 k ! · n ( n - 1 ) ··· ( n - k + 1 ) n k · ( np ) k · q n · q - k 3 / 13 Xinghua Zheng Lect 11: The Poisson distribution

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MOMENTS AND FACTORIAL MOMENTS C EXAMPLE: COUNTING DEATHS, ctd P [ X = k ] = 1 k !
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lect11 - MOMENTS AND FACTORIAL MOMENTS THE POISSON...

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