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Poisson Approximation

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

bdpb \rightarrow^d p if n>>1n >> 1, p<<1p << 1 and λ=np\lambda = np

In Probability,

(nx)=n!x!(nx)!{n \choose x} = {n! \over x! (n-x)!} b(n,p)=Poisson(λ) where λ=npb(n, p) = \text{Poisson}(\lambda) ~ \text{where} ~ \lambda = np n1 and p1n \gg 1 ~\text{and} ~ p \ll 1

approximates binomial distribution.

bp    n1 & p1 & λ=npb \rightarrow p \iff n \gg 1 ~ \& ~ p \ll 1 ~ \& ~ \lambda = np

where bb is binomial and pp is poisson