Proof that 2 geometric random variable is NB

Question

Can someone write me the proof of sum of two geometric variable are negative binomial ? $X\sim \mathrm{Geo}(p)$ and $Y\sim \mathrm{Geo}(p)$ how can i proof that $Z=X+Y \sim \mathrm{NegBin}(2,p)$?

Answer 1

Under the assumption that $X$ and $Y$ are independent random variables with support $k\in\{0,1,2,3\}$ failures, we have the probability that random variable $Z$ equals integer $m$ being $$\begin{align}P(Z=m)&=\sum_{k=0}^mP(X=k)P(Y=m-k)\\&=\sum_{k=0}^m(1-p)^kp\times(1-p)^{m-k}p\\&=(m+1)p^2(1-p)^m\\&={m+2-1 \choose 2-1}p^2(1-p)^m\sim NB(2,p) \end{align}$$