The Recursive Property of the Binomial Distribution Background

The Recursive Property of the Binomial Distribution Background Historically, the Binomial Distribution evaluation has been subject to approximation due to the laborious math involved. When the product np is less than five, a Poisson approximation is used. When np is equal or greater than five a normal approximation is used.

Derivation The Binomial probability of exactly x occurrences in a sample of n when the probability of an occurrence is p per unit within the sample is x n-x p(x) = (n! /(x!(n-x)!)) p (1 -p). The probability of zero occurrences is then equal n p(0) = (1 -p) .

Derivation Continued The probability of one occurrence is then equal n-1 n p(1) = np(1 -p) = n(p/(1 -p)) [ (1 -p) ] = n{p/(1 -p)} [ p(o) ]. The probability of two occurrences is then equal 2 n-2 p(2) = [n(n-1)/2] p ( 1 -p) n-1 = {(n-1)/2}{(p/(1 -p)} [ np(1 -p) ]

Derivation Continued p(2) = {(n-1)/2}{p/(1 -p)} [ p(1) ]. The probability of three occurrences is then equal 3 n-3 p(3) = [n(n-1)(n-2)/((2)(3))] p ( 1 -p) 2 n-2 = {(n-2)/3}{(p/(1 -p)} [ (n(n-1)/2)p (1 -p) ] = {(n-2)/3}{p/(1 -p)} [ p(2) ].

The Recursive Expression Generalizing, from the above probabilities, the Binomial recursive expression becomes p(x) = [ (n+1 -x) / x ] [ p/(1 -p) ] p(x-1).