정규 ⇒ 카이제곱

\begin{align*} &M_Y(t) = E(e^{tY}) = E(e^{tX^2}) \\[15pt] &= \int_{-\infty}^{\infty} e^{tX^2} \cdot \frac{1}{\sqrt{2\pi}} \exp\bigg({-\frac{x^2}{2}}\bigg) \\[15pt] &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \exp\bigg[ -{\frac{x^2}{2}}~(1-2t)\bigg] \\[15pt] &= \frac{1}{\sqrt{2\pi}}~\cdot~ \sqrt{2\pi}~~\cdot~~ \sqrt{\frac{1}{1-2t}} \\[15pt] &= (1-2t) \raisebox{0.7em} {$ \scriptstyle - \textstyle \frac{1}{2}$} \end{align*}