正規分布の積率母関数の導出mX(t)=E(etX)=∫−∞∞etxf(x)dx=∫−∞∞12πσ2exp[−(x−μ)22σ2+tx]dx=∫−∞∞12πσ2exp[−12σ2(x2−2μx+μ2−2σ2tx)]dx=∫−∞∞12πσ2exp[−12σ2((x−(μ+σ2t))2+2μσ2t+σ4t2)]dx=∫−∞∞12πσ2exp[−(x−(μ+σ2t))22σ2+μt+σ2t22]dx=eμt+σ2t22∫−∞∞12πσ2exp[−(x−(μ+σ2t))22σ2]dx=eμt+σ2t22\begin{equation*}\begin{split}m_X(t)&=E(\mathrm{e}^{tX})\\ &=\displaystyle \int_{ - \infty }^{ \infty }\mathrm{e}^{tx}f(x)dx\\ &=\displaystyle \in