\[\begin{align*} E(X) &= \sum_{k=0}^n k\binom{n}{k} p^kq^{n-k}\\ &= \sum_{k=1}^n k\binom{n}{k} p^kq^{n-k}\\ &= \sum_{k=1}^n n\binom{n-1}{k-1} p^kq^{n-k}\\ &= np\sum_{k=1}^n \binom{n-1}{k-1} p^{k-1} q^{(n-1)-(k-1)}\\ &= np\sum_{j=0}^m\binom{m}{j}p^jq^{m-j} \\ &= np \end{align*}\]
\[\begin{align*} E(X^2) &= \sum_{k \ge0}^n k^2\binom{n}{k} p^k q^{n-k}\\ &= \sum_{k=0}^n kn\binom{n-1}{k-1}p^k q^{n-k} \\ &= np \sum_{k=1}^n k \binom{n-1}{k-1} p^{k-1} q^{(n-1) - (k-1)} \\ &= np \sum_{j=0}^m (j+1) \binom{m}{j} p^j q^{m-j} \\ &= np \left(\sum_{j=0}^m j \binom{m}{j} p^j q^{m-j} + \sum_{j=0}^m \binom{m}{j} p^j q^{m-j}\right) \\ &= np \left(\sum_{j=0}^m m \binom{m-1}{j-1} p^{j} q^{m-j} + \sum_{j=0}^m \binom{m}{j} p^j q^{m-j}\right) \\ &= np \left((n-1)p \sum_{j=1}^m \binom{m-1}{j-1} p^{j-1} q^{(m-1)-(j-1)} + \sum_{j=0}^m \binom{m}{j} p^j q^{m-j}\right) \\ &= np\left((n-1)p \sum_{j=1}^m \binom{m-1}{j-1} p^{j-1} q^{(m-1)-(j-1)} + \sum_{j=0}^m \binom{m}{j} p^j q^{m-j} \right) \\ &= np\left((n-1)p(p+q)^{m-1} + (p+q)^m \right) \\ &= np\left((n-1)p+1\right)\\ &= n^2p^2 + np(1-p) \\ Var(X) &= E(X^2) - [E(X)]^2 \\ &= np(1-p) + n^2p^2 - (np)^2 \\ &= np(1-p) \\ &= npq \end{align*}\]
\[\begin{align*} M_X(t) &= E(\exp\left(tX\right)) \\ &= \sum_{k=0}^n Pr(X=k)e^{tk} \\ &= \sum_{k=0}^n \binom{n}{k} p^k(1-p)^{n-k}e^{tk} \\ &= \sum_{k=0}^n \binom{n}{k} (pe^t)^k(1-p)^{n-k}\\ &= (1-p+pe^t)^n \\ &= (q+pe^t)^n \end{align*}\]