## Info

y.yJ y!( n - y)! The probability of obtaining y bacteria in any sample is then n! fXy

The previous equation is called the Bernoulli distribution, from which we will derive the Poisson distribution.

n! may be expanded as n(n - 1)(n - 2)(n - 3)...(n - y + 1)(n - y)!. Using this new expression, Equation (2.30) may be rewritten as

Prob( y) = ^T n{n-1Xn-2Xn-3)...{n-y+1)(n-y)!-y^ {n - y)!Ny

where (n - y)! will cancel out from both the numerator and the denominator of the second factor on the right. n in the numerator may be factored out to produce, after canceling out (n - y)!.

When n! is expanded as n(n - 1)(n - 2)(n - 3).{n - y + 1){n - y)!, the number of factors in this expansion is y + 1. This is shown as follows: (n - 1)(n - 2)(n - 3). {n - y + 1){n - y)! has y factors. Including the n, the total number of factors is therefore y + 1. Because {n - y)! was canceled out from both the numerator and the y

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