Poisson Approximation

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$P (\lambda) = {{e^{-\lambda} \lambda^x} \over x!}$

$b \rightarrow^d p$ if $n >> 1$ , $p << 1$ and $\lambda = np$

In Probability,

{n \choose x} = {n! \over x! (n-x)!}

b(n, p) = \text{Poisson}(\lambda) ~ \text{where} ~ \lambda = np

n \gg 1 ~\text{and} ~ p \ll 1

approximates binomial distribution.

b \rightarrow p \iff n \gg 1 ~ \& ~ p \ll 1 ~ \& ~ \lambda = np

where $b$ is binomial and $p$ is poisson

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