\begin{align}
P(X=k) = \begin{cases} _{n}C_k p^k(1-p)^{n-k} & (k=0,1,\cdots,n) \\
0&(\text{others})
\end{cases}
\end{align}

\begin{align}
E(X) &= \sum_{k=0}^n k \times P(X=k)\\
&=\sum_{k=0}^n k \times _{n}C_k p^k(1-p)^{n-k}&&\\
&=\sum_{k=1}^n \cfrac{n!}{(k-1)!(n-k)!}p^k (1-p)^{n-k}\\
&(\text{Put } t \text{ as } k-1)\\
&=np \underline{\sum_{t=0}^{n-1} \cfrac{(n-1)!}{t!((n-1)-t)!} p^t (1-p)^{(n-1)-t}}\\
&=np
\end{align}

\begin{align}
E(X^2) &= \sum_{k=0}^n k^2 \times P(X=k)\\
&=\sum_{k=0}^n k^2 \times _{n}C_k p^k(1-p)^{n-k}&&\\
&=\sum_{k=1}^n \cfrac{k\cdot n!}{(k-1)!(n-k)!}p^k (1-p)^{n-k}\\
&(\text{Put } t \text{ as } k-1)\\
&=np \sum_{t=0}^{n-1} (t+1)\cfrac{(n-1)!}{t!((n-1)-t)!} p^t (1-p)^{(n-1)-t}\\
&(\text{Put } Y \sim B(n-1,p) )\\
&=np\left(E(Y) + \sum_{t=0}^{n-1}P(Y=t)\right)\\
&=np((n-1)p + 1)\\
&=(np)^2 + np(1-p)
\end{align}

\begin{align}
V(X) &= E(X^2) – E(X)^2\\
&= (np)^2 + np(1-p) – (np)^2\\
&= np(1-p)
\end{align}

\begin{align}
M_X(t) &= E\left(e^{tX}\right) \\
&= \sum_{k=0}^n e^{tk} {}_nC_k p^k (1-p)^{n-k}\\
&= \sum_{k=0}^n {}_nC_k (pe^{t})^k (1-p)^{n-k}\\
&=(pe^t +1 – p)^n
\end{align}

\begin{align}
E(X) &= M_X'(0)\\
&=n(pe^t + 1-p)^{n-1} \cdot pe^t |_{t=0}\\
&=np\\
E(X^2) &= M_X”(0)\\
&=n(n-1)(pe^t + 1-p)^{n-2} \cdot (pe^t)^2 \\
&\qquad + n(pe^t+1-p)^{n-1}\cdot pe^t |_{t=0}\\
&=(np)^2 + np(1-p)
\end{align}