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Unformatted text preview: N O T E S O N P R O B A B I L I T Y P A U L M O O N N Y 2 8 Chapter 4 Poisson Distribution The Poisson distribution is the limit of the binomial distribution as N ! 1 and p ! under the condition that the mean & = Np (of the binomial distribution) remains constant. It turns out that the Poisson distribution provides a good approximation to the binomial distribution even in cases where N is only moderately large (e.g., N = 100 ) as long as p is su¢ ciently small. 4.1 Derivation Recall the binomial distribution P [ X = k ] = & N k ¡ p k q N & k (4.1) from Theorem 30 which we rewrite as P [ X = k j N;p ] = ¢ N ( N & 1) ( N & 2) ::: ( N & k + 1) k ! £ p k (1 & p ) N & k : (4.2) The notation P [ X = k j N;p ] means that we consider P [ X = k ] for temporarily &xed N and p which are otherwise arbitrary. We rewrite equation (4.2) as P [ X = k j N;p ] = ¢ & k k ! £¢& N N ¡& N & 1 N ¡ ::: & N & k + 1 N ¡£ " & 1 & & N ¡ & k #" & 1 & & N ¡ N # ; (4.3) where we have used p = & N ; (1 & p ) N & k = h...
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This note was uploaded on 12/10/2009 for the course ECE el6303 taught by Professor Moon during the Spring '09 term at NYU Poly.
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