Definition (Binomial RV)

Let $X$ be a RV. The number of successes $X$ in $n$ independent Bernoulli trials with probability $p$ of success. This is called a binomial RV with parameters $(n,p)$ with pmf $$p(k)={n\choose k}p^{k}(1-p)^{n-k},\ k\in\mathcal{X}=\{0,1,\cdots,n\}$$For RV $X\sim \mbox{Binomial}(n,p)$ $$\begin{align*} &E[X]=np \\ &\mbox{Var}(X)=np(1-p) \end{align*}$$