베르누이 분포

\begin{align*} E(X) &= \displaystyle\sum_{x}~xP(X=x) \\[15pt] &= \displaystyle\sum_{x=0}^{1}~ p^x(1-p)^{1-x}\;~I(x \in \{0, 1\}) \\[15pt] &= 0 + 1~\cdot~p^{1}(1-p)^{1-1} \\[10pt] &= p \end{align*}

\begin{align*} E(X^2) &= \displaystyle\sum_{x}~x^2P(X=x) \\[15pt] &= \displaystyle\sum_{x}~x^2 \cdot p^x(1-p)^{1-x}~ ~I\big(x \in \{0, 1\}\big) \\[15pt] &= p \end{align*}

\begin{align*} Var(X) &= E(X-E(X))^2 = \sum_{x}~\{x - E(X)\}^2~P(X=x) \\[15pt] &= \sum_{x=0}^{1}~\{x-E(X)\}^2~ p^x~(1-p)^{1-x} \\[15pt] &= \sum_{x=0}^{1}~\{x-p\}^2~ p^x~(1-p)^{1-x} \\[15pt] &= p^2\cdot p^0\cdot(1-p)^{1-0}~+~ (1-p)^{2}\cdot p^0\cdot(1-p)^{1-1} \\[15pt] &= p^2\cdot(1-p) + (1-p)^{2}\cdot p \\[15pt] &= p~(1-p) \end{align*}

\begin{align*} Var(X) &= E(X^2) - \{E(X)\}^2 = p-p^2 = p(1-p) \end{align*}

\begin{align*} M_X(t) &= \sum_{x}~e^{tx}~P(X=x) = \sum_{x=0}^{1} ~e^{tx}~p^x(1-p)^{1-x} \\[15pt] &= 1-p + e^{t}p \end{align*}

\dfrac{d}{dt}~ M_X(t)~\biggr\rvert_{t = 0}~ =~ pe^t~\biggr\rvert_{t = 0}~ =~ p~ =~ E(X) \\[20pt] \dfrac{d^2}{d^2t}~ M_X(t)~\biggr\rvert_{t = 0}~ =~ pe^t~\biggr\rvert_{t = 0}~ =~ p~ =~ E(X^2)