Turns out for large \(n\) and small \(p\), a binomial distribution can be approximated by a Poisson distribution with \(\lambda = np\). Why ?

\[ \begin{aligned} {n \choose k} p^k (1-p)^{n-k} &= {n! \over (n-k)!\, k!} ({\lambda \over n})^k (1 - {\lambda \over n})^{n-k} \\ &= {n(n-1) \cdots (n-k+1) \over n^k} {\lambda^k \over k!} {\color{blue}{(1 - {\lambda \over n})^n} \over (1 - {\lambda \over n})^k} \\ &= \color{blue}{e^{-\lambda}} {\lambda^k \over k!} & \text{as }n \rightarrow \infty \\ \end{aligned} \]

which is the Poisson distribution !

\(\Box\)

Source: Kiryl Tsishchanka

# posted by rot13(Unafba Pune) @ 7:17 PM