正規分布(normal distribution)の積率母凾数(moment-generating function)と期待値(expected value)・分散(variance)

正規分布

$$\begin{array}{rcl}N(\mu ,{\sigma }^2)& =& \frac{1}{\sqrt{2\pi {\sigma }^2}}{\mathrm{e}}^{\frac{-(x-\mu {)}^2}{2{\sigma }^2}}\end{array}$$

積率母凾数

$$\begin{array}{rcl}{\displaystyle M_X(t)}& \equiv & {\displaystyle E[{\mathrm{e}}^{tX}]}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}({\mathrm{e}}^{tx})\frac{1}{\sqrt{2\pi {\sigma }^2}}{\mathrm{e}}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}\mathrm{d}x}\\ & =& {\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}({\mathrm{e}}^{tx}){\mathrm{e}}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}\mathrm{d}x}\\ & =& {\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}+tx}\mathrm{d}x}\\ & =& {\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{(x-\mu {)}^2+(2{\sigma }^2)(tx)}{2{\sigma }^2}}\mathrm{d}x}\\ & =& {\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{x^2-2x\mu +{\mu }^2-2x{\sigma }^{2}t}{2{\sigma }^2}}\mathrm{d}x}\\ & =& {\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle {\mathrm{e}}^{-\frac{x^2-2x\mu +{\mu }^2-2x{\sigma }^{2}t}{2{\sigma }^2}-\frac{2\mu {\sigma }^{2}t+{\sigma }^4{t}^2}{2{\sigma }^2}+\frac{2\mu {\sigma }^{2}t+{\sigma }^4{t}^2}{2{\sigma }^2}}{\displaystyle \mathrm{d}x}}}\\ & =& {\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle {\mathrm{e}}^{-\frac{x^2-2x\mu +{\mu }^2-2x{\sigma }^{2}t+2\mu {\sigma }^{2}t+{\sigma }^4{t}^2}{2{\sigma }^2}+\frac{2\mu {\sigma }^{2}t+{\sigma }^4{t}^2}{2{\sigma }^2}}{\displaystyle \mathrm{d}x}}}\\ & =& {\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle {\mathrm{e}}^{-\frac{(x-\mu -{\sigma }^{2}t{)}^2}{2{\sigma }^2}+\frac{2\mu {\sigma }^{2}t+{\sigma }^4{t}^2}{2{\sigma }^2}}{\displaystyle \mathrm{d}x\dots a^2-2ab+b^2-2ac+2bc+c^2=(a-b-c{)}^2}}}\\ & =& {\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle {\mathrm{e}}^{-\frac{(x-\mu -{\sigma }^{2}t{)}^2}{2{\sigma }^2}+(\mu t+\frac{{\sigma }^2{t}^2}{2})}{\displaystyle \mathrm{d}x}}}\\ & =& {\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{(x-\mu -{\sigma }^{2}t{)}^2}{2{\sigma }^2}}{\displaystyle {\mathrm{e}}^{(\mu t+\frac{{\sigma }^2{t}^2}{2})}{\displaystyle \mathrm{d}x}}}}\\ & =& {\displaystyle {\mathrm{e}}^{(\mu t+\frac{{\sigma }^2{t}^2}{2})}{\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{(x-\mu -{\sigma }^{2}t{)}^2}{2{\sigma }^2}}\mathrm{d}x}}}\\ & =& {\displaystyle {\mathrm{e}}^{(\mu t+\frac{{\sigma }^2{t}^2}{2})}\dots \frac{1}{\sqrt{2\pi {\sigma }^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{(x-\mu -{\sigma }^{2}t{)}^2}{2{\sigma }^2}}\mathrm{d}x=N(\mu +{\sigma }^{2}t,{\sigma }^2)\mathrm{の}\mathrm{総}\mathrm{和}=1}\end{array}$$

期待値・分散

$$\begin{array}{rcl}{\displaystyle M_X^{(m)}(0)}& \equiv & \frac{{\mathrm{d}}^m}{{\mathrm{d}}^{m}t}{M}_x(t){|}_{t=0}\\ & =& {\displaystyle E[X^m{\mathrm{e}}^{tX}]{|}_{t=0}}\\ & =& {\displaystyle E[X^m]}\end{array}$$ $$\begin{array}{rcl}{\displaystyle E[X]}& =& {\displaystyle M_X^{(1)}(0)}\\ & =& {\displaystyle \left\{\frac{\mathrm{d}}{\mathrm{d}t}\left({\mathrm{e}}^{(\mu t+\frac{{\sigma }^2{t}^2}{2})}\right)\right\}{|}_{t=0}}\\ & =& {\displaystyle \{{\mathrm{e}}^{(\mu t+\frac{{\sigma }^2{t}^2}{2})}(\mu +{\sigma }^{2}t)\}{|}_{t=0}}\\ & =& {\displaystyle {\mathrm{e}}^{(\mu 0+\frac{{\sigma }^2{0}^2}{2})}(\mu +{\sigma }^{2}0)}\\ & =& {\displaystyle {\mathrm{e}}^0(\mu +0)}\\ & =& {\displaystyle \mu }\end{array}$$ $$\begin{array}{rcl}{\displaystyle E[X^2]}& =& {\displaystyle M_X^{(2)}(0)}\\ & =& {\displaystyle \left\{\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}\left({\mathrm{e}}^{(\mu t+\frac{{\sigma }^2{t}^2}{2})}\right)\right\}{|}_{t=0}}\\ & =& {\displaystyle \left\{\frac{\mathrm{d}}{\mathrm{d}t}({\mathrm{e}}^{(\mu t+\frac{{\sigma }^2{t}^2}{2})}(\mu +{\sigma }^{2}t))\right\}{|}_{t=0}}\\ & =& {\displaystyle \left[{\displaystyle \left\{\frac{\mathrm{d}}{\mathrm{d}t}\left({\mathrm{e}}^{(\mu t+\frac{{\sigma }^2{t}^2}{2})}\right)\right\}(\mu +{\sigma }^{2}t){\displaystyle +\left({\mathrm{e}}^{(\mu t+\frac{{\sigma }^2{t}^2}{2})}\right)\left\{\frac{\mathrm{d}}{\mathrm{d}t}(\mu +{\sigma }^{2}t)\right\}}}\right]{|}_{t=0}}\\ & =& {\displaystyle \left[{\displaystyle {\mathrm{e}}^{(\mu t+\frac{{\sigma }^2{t}^2}{2})}{(\mu +{\sigma }^{2}t)}^2{\displaystyle +{\mathrm{e}}^{(\mu t+\frac{{\sigma }^2{t}^2}{2})}{\sigma }^2{\displaystyle }}}\right]{|}_{t=0}}\\ & =& {\displaystyle {\mathrm{e}}^{(\mu 0+\frac{{\sigma }^2{0}^2}{2})}{(\mu +{\sigma }^{2}0)}^2{\displaystyle +{\mathrm{e}}^{(\mu 0+\frac{{\sigma }^2{0}^2}{2})}{\sigma }^2}}\\ & =& {\displaystyle {\mu }^2+{\sigma }^2}\end{array}$$ $$\begin{array}{rcl}V[X]& =& E[X^2]-E[X{]}^2\\ & =& {\mu }^2+{\sigma }^2-{\mu }^2\\ & =& {\sigma }^2\end{array}$$