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Chapter 3 Random Variables Appendix (Poisson Distribution)

Chapter 3 Random Variables Appendix (Poisson Distribution)...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1306 INTRODUCTORY STATISTICS (SEMESTER 1 2008/2009) Poisson Distribution Indeed, Poisson distribution can be considered as a limiting case of binomial distribution. Let Y be the number of events (S) occurred within a time interval and the probability that an event occurs is p . Furthermore, each trial is independent from each other. Then, out of n possible events within the time interval, the probability that y events occur is P ( Y = y ) = n y ± p y (1 p ) n y = n ! y ! ( n y )! p y (1 p ) n y : Denote p = . Intuitively, is rate of occurrence within a time interval and equals np , the mean number of events. The probability can be written as P ( Y = y ) = n ! y ! ( n y )! n ± y 1 n ± n y = n ! n y ( n y )! y y ! 1 n ± n 1 n ± y : As n ! 1 , lim n !1 1 n ± n = e and lim n !1 1 n ± y = 1 since n ! 0 . Furthermore, the limit of n ! n y ( n y )! log n ! n y ( n y )! = log n ! y log n log( n y )! : Secondly, by Stiring±s formula, i.e. lim n !1 n ! p 2 ±n ² n e ³ n = 1 or lim n !1 e n n ! n n + 1 2 = p 2 ±; we have lim n !1 log n ! = log p 2 ± + lim n !1 [ n log n n ] 1
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Unformatted text preview: n !1 log( n & y )! = log p 2 & + lim n !1 [( n & y ) log ( n & y ) & ( n & y )] : Then, lim n !1 log n ! n y ( n & y )! = lim n !1 [ n log n & n ] & y lim n !1 log n & lim n !1 [( n & y ) log ( n & y ) & ( n & y )] = lim n !1 & ( n & y ) log ± n n & y ²³ & y = lim n !1 & ´ 1 & y n µ n log ± n n & y ²³ & y = lim n !1 & ´ 1 & y n µ log ± n n & y ² n ³ & y = & lim n !1 ´ 1 & y n µ lim n !1 log ´ 1 & y n µ & n & y = ( & 1) ± ( & y ) & y = : Therefore, lim n !1 n ! n y ( n & y )! = 1 : Consequently, lim n !1 P ( Y = y ) = lim n !1 " n ! n y ( n & y )! ± ± y y ! ²± 1 & ± n ² n ± 1 & ± n ² & y # = 1 ± ± ± y y ! ² e & & ± 1 = e & & ± y y ! . As a result, the Poisson distribution can be considered as the limiting case of Bino-mial distribution as n ! 1 . 2...
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