If X∼Binomial(n,p)X \sim \text{Binomial}(n, p)X∼Binomial(n,p), find P(X=k)P(X=k)P(X=k) as n→∞,p→0n \to \infty, p \to 0n→∞,p→0 such that np=λnp = \lambdanp=λ.
e−λλkk!\frac{e^{-\lambda} \lambda^k}{k!}k!e−λλk
12πλe−(k−λ)22λ\frac{1}{\sqrt{2\pi\lambda}} e^{-\frac{(k-\lambda)^2}{2\lambda}}2πλ1e−2λ(k−λ)2
(nk)pk(1−p)n−k\binom{n}{k} p^k (1-p)^{n-k}(kn)pk(1−p)n−k
e−λe^{-\lambda}e−λ