Definition (Negative Binomial RV)

Let $X$ be a discrete RV with range $\mathcal{X}=\{k,k+1,\cdots\}$, for $k\ge1$. The number of Bernoulli trials needed for $k$ successes is called a negative RV with parameters $(k,p)$ and pmf $$p(n)={n-1\choose k-1}p^{k}(1-p)^{n-k}$$For RV $X\sim NB(k,p)$ $$\begin{align*} E[X]&=\frac{k}{p}\\ \mbox{Var}(X)&=\frac{k(1-p)}{{p^{2}}} \end{align*}$$