> [!NOTE] Theorem > Let $n\in \mathbb{N}$ and $p\in[0,1].$ Let $X$ be a [[Discrete random variables|discrete real-valued random]] that has a [[Binomial Distribution|binomial distribution]] with parameters $n$ and $p.$ Then $X$ is [[Square-Integrable Discrete Real-Valued Random Variable|square-integrable]] and its [[Variance of Square-Integrable Discrete Real-Valued Random Variable|variance]] is $np(1-p).$ **Proof**: By [[Expectation of Real-Valued Function of Discrete Real-Valued Random Variable]], $\begin{align} \mathbb{E}[X^{2}]&=\sum_{k=0}^{\infty} p_{X}(k)\cdot k^{2} = \sum_{k=0}^{\infty} k^{2} {n \choose k} p^{k} (1-p)^{n-k} \\ &=np \sum_{k=1}^{\infty} k {n-1 \choose k-1} p^{k-1}(1-p)^{(n-1)-(k-1)} \end{align}$using [[Factors of Binomial Coefficient]]. Now $\begin{aligned} \mathbb{E}[X^{2}]&=np\sum_{j=0}^{m}\left(j+1\right)\binom{m}{j}p^{j}q^{m-j} \\ &=np\left(\sum_{j=0}^mj\binom mjp^jq^{m-j}+\sum_{j=0}^m\binom mjp^jq^{m-j}\right) \\ &=np\left(\sum_{j=0}^mm\binom{m-1}{j-1}p^jq^{m-j}+\sum_{j=0}^m\binom mjp^jq^{m-j}\right) \\&\text{=} np\left((n-1)p\sum_{j=1}^m\binom{m-1}{j-1}p^{j-1}q^{(m-1)-(j-1)}+\sum_{j=0}^m\binom mjp^jq^{m-j}\right) \\ &= np\left((n-1)p(p+q)^{m-1}+(p+q)^m\right) \\ &= np\left(\left(n-1\right)p+1\right) \\ &= n^{2}p^{2}+np(1-p) \end{aligned}$ Therefore $\text{Var}(X)=\mathbb{E}[X^{2}]-(\mathbb{E}[X])^{2} = n^{2}p^{2}+np(1-p)-n^{2}p^{2} = np(1-p)$using [[Expectation of Binomial Distribution]].