04_04 - p 4-31 Note For X~binomial(n p where(i n large(ii p...

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p. 4-31 Poisson Process Example: (1) # of earthquakes occurring during some fixed time span (2) # of people entering a bank during a time period To model them, we can Divide the time period, say [0, t ], into n small intervals Note : For X ~binomial( n , p ), where (i) n large; (ii) p small, distribution of X Poisson(  np ) E ( X ) np mean of the Poisson   Var (X) np (1 p ) variance of the Poisson   Make the intervals so small (i.e., n is large) that at most one event can occur in each interval p. 4-32 Definition. A Poisson process with rate is a family of r.v.’s N t , 0 t < , for which N 0 = 0 and N t N s ~ Poisson( · ( t s )), for 0 s < t < , and Assume that the number of events to occur in non-overlapping intervals are independent are independent whenever 0 s 1 < t 1 s 2 < t 2 s m < t m . N t i N s i ,i =1 , 2 ,...,m Then, we can treat the number of events in a single interval as a Bernoulli r.v. with a small p n
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This note was uploaded on 02/15/2012 for the course MATH 2810 taught by Professor Shao-weicheng during the Fall '11 term at National Tsing Hua University, China.

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04_04 - p 4-31 Note For X~binomial(n p where(i n large(ii p...

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