> [!NOTE] Theorem > Let $n\in \mathbb{N}$ and $p=[0,1].$ Let $X$ be a [[Discrete random variables|discrete real-valued random]] that has a [[Binomial Distribution|binomial distribution]] with parameters $n$ and $p.$ Let $\lambda=np.$ Then, when $n$ is large, $X$ can be approximated by a [[Poisson Distribution|poisson distribution]] with parameter $\lambda$: that is $\lim_{ n \to \infty } {n \choose x} p^{k}(1-p)^{n-k} = \frac{\lambda^{k}}{k!}e^{-\lambda} $ **Proof**: Let $X\sim \text{B}(n,p)$ and $\lambda=np.$ Let $p_{X}$ denote the [[Probability Mass Function of Discrete Real-Valued Random Variable|PMF]] of $X.$ As $n\to \infty,$ we have $\begin{align} p_{X}(k) &= {n \choose k} p^{k}(1-p)^{n-k} \\ &= \frac{n!}{k!(n-k)!} \left( \frac{\lambda}{n} \right)^{k}\left( 1- \frac{\lambda}{n} \right)^{n-k} \\ &= \frac{\lambda^{k}}{k!} \frac{n(n-1)\cdots (n-k-1)}{n^{k}} \left( 1- \frac{\lambda}{n} \right)^{n} \left( 1- \frac{\lambda}{n} \right)^{-k} \\ &\to \frac{\lambda^{k}}{k!}e^{-\lambda} \end{align}$using [[Equivalence of Real Exponential Function as Limit of a Sequence and Power Series]] and $n(n-1)...(n-k+1)/n^k\to1,\quad\left(1-\frac\lambda n\right)^n\to e^{-\lambda},\quad\frac\lambda n\to0.$