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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 Binomial distribution as n ! 1 . 2...
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 Fall '08
 Prof
 Statistics, Binomial, Normal Distribution, Poisson Distribution, Mean, Probability theory, Binomial distribution, lim

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