\begin{align}
P(X=k) = \begin{cases} \binom{r+k-1}{k} p^r (1-p)^{k} & (k=0,1,\cdots) \\
0&(\text{others})
\end{cases}
\end{align}

\begin{align}
E(X) &= \sum_{k=0}^n k \times P(X=k)\\
&=\sum_{k=0}^\infty k \times \binom{r+k-1}{k}p^r q^k&&\\
&=\sum_{k=1}^\infty k \times \cfrac{(r+k-1)!}{k!(r-1)!}p^r q^k&&\\
&=\sum_{k=1}^\infty \cfrac{(r+k-1)!}{(k-1)!r!}p^{r+1} q^{k-1} \cdot \frac{rq}{p}&&\\
&(\text{Put } t \text{ as } k-1)\\
&=\frac{rq}{p}\cdot\underline{ \sum_{t=0}^\infty \cfrac{((r+1)+t-1)!}{t!((r+1)-1)!}p^{r+1} q^{t}}&&\\
&=\frac{rq}{p}
\end{align}

\begin{align}
E(X^2) &= \sum_{k=0}^n k^2 \times P(X=k)\\
&(\text{Put } t \text{ as } k-1)\\
&=\frac{rq}{p}\cdot \sum_{t=0}^\infty (t+1)\cdot\cfrac{((r+1)+t-1)!}{t!((r+1)-1)!}p^{r+1} q^{t}&&\\
&=\frac{rq}{p}\left(E(Y)+1\right)\\
&=\frac{rq}{p}\left(\frac{(r+1)q}{p}+1\right)\\
&=\frac{rq(rq+1)}{p^2}
\end{align}

\begin{align}
V(X) &= E(X^2) – E(X)^2\\
&= \frac{rq(rq+1)}{p^2} – \left(\frac{rq}{p}\right)^2\\
&= \frac{rq}{p^2}
\end{align}

\begin{align}
M_X(t) &= E\left(e^{tX}\right) \\
&= \sum_{k=0}^\infty e^{tk} \binom{r+k-1}{k}p^r q^k\\
&= \left(\frac{p}{1-e^tq}\right)^r \underline{\sum_{k=0}^\infty \binom{r+k-1}{k} (1-e^tq)^r(e^tq)^k}\\
&= \left(\frac{p}{1-e^tq}\right)^r
\end{align}

\begin{align} E(X) &= M_X'(0)\\ &=\left.r\left(\frac{p}{1-e^tq}\right)^{r-1} \frac{e^tpq}{(1-e^tq)^2} \right|_{t=0}\\ &=\left.rM_X(t) \cdot \frac{e^tq}{1-e^tq} \right|_{t=0}\\ &=\frac{rq}{p}\\ E(X^2) &= M_X''(0)\\ &=\left.rM_X'(t) \cdot \frac{e^tq}{1-e^tq} \right|_{t=0} \\ &\qquad +\left. rM_X(t)\cdot \frac{e^tq(1-e^tq)+(e^tq)^2}{(1-e^tq)^2} \right|_{t=0}\\ &=\left(\frac{rq}{p}\right)^2 + \frac{rq}{p^2} \end{align}