\begin{align}
P(X=k) = \begin{cases} \cfrac{\lambda^k}{k!}e^{-\lambda} & (k=0,1,\cdots) \\
0&(\text{others})
\end{cases}
\end{align}

\begin{align}
E(X) &= \sum_{k=0}^\infty k \times P(X=k)\\
&=\sum_{k=0}^\infty k \times \cfrac{\lambda^k}{k!}e^{-\lambda}&&\\
&=\sum_{k=1}^\infty \cfrac{\lambda \cdot \lambda^{k-1}}{(k-1)!}e^{-\lambda}\\
&(\text{Put } t \text{ as } k-1)\\
&=\lambda \underline{\sum_{t=0}^\infty \cfrac{\lambda^{t}}{t!}e^{-\lambda}}\\
&=\lambda
\end{align}

\begin{align}
E(X^2) &= \sum_{k=0}^\infty k^2 \times P(X=k)\\
&=\sum_{k=0}^\infty k^2 \times \cfrac{\lambda^k}{k!}e^{-\lambda}&&\\
&=\sum_{k=1}^\infty k \times \cfrac{\lambda \cdot \lambda^{k-1}}{(k-1)!}e^{-\lambda}\\
&(\text{Put } t \text{ as } k-1)\\
&=\lambda \sum_{t=0}^\infty (t+1)\cfrac{\lambda^{t}}{t!}e^{-\lambda}\\
&=\lambda(E(X) + 1)\\
&=\lambda^2 + \lambda
\end{align}

\begin{align}
V(X) &= E(X^2) – E(X)^2\\
&= \lambda^2 + \lambda – \lambda^2\\
&= \lambda
\end{align}

\begin{align}
M_X(t) &= E\left(e^{tX}\right) \\
&= \sum_{k=0}^\infty e^{tk} \cfrac{\lambda^k}{k!}e^{-\lambda}\\
&= \underline{\sum_{k=0}^\infty \cfrac{(e^t\lambda)^k}{k!} \exp(-e^t\lambda)}\times \cfrac{\exp(-\lambda)}{\exp(-e^t\lambda)}\\
&= \cfrac{\exp(-\lambda)}{\exp(-e^t\lambda)}\\
&=\exp(\lambda(e^t-1))
\end{align}

\begin{align}
E(X) &= M_X'(0)\\
&=\exp(\lambda(e^t-1))\cdot \lambda e^t |_{t=0}\\
&=\lambda\\
E(X^2) &= M_X”(0)\\
&= \exp(\lambda(e^t-1))\cdot (\lambda e^t)^2\\
&\qquad + \exp(\lambda(e^t-1))\cdot \lambda e^t |_{t=0}\\
&=\lambda^2 + \lambda
\end{align}