Definition (Geometric RV)

Let $X$ be a discrete RV with range $\mathcal{X}=\{1,2,\cdots\}$. The number of fails $X$ until a first success is called a geometric RV with parameter $p$ and pmf $$p(n)=(1-p)^{n-1}p, \ n\in\mathcal{X}=\{1,2,\cdots\}$$For RV $X\sim \mbox{Geometric}(p)$ $$\begin{align*} E[X]&=\frac{1}{p}\\ \mbox{Var}(X)&=\frac{1-p}{p^{2}} \end{align*}$$