単回帰モデルの最小二乗推定量の分布

単回帰モデルの最小二乗推定量$\hat{\alpha },\hat{\beta }$の分布

単回帰モデル

$$\begin{array}{rcl}{y}_i& =& \alpha +\beta x_i+{ϵ}_i(i=1,\cdots ,n)\dots {ϵ}_i\stackrel{iid}{\sim }\mathrm{N}(0,{\sigma }^2)\\ \mathrm{E}\left[y_i\right]& =& \mathrm{E}[\alpha +\beta x_i+{ϵ}_i]\\ & =& \alpha +\beta x_i+\mathrm{E}\left[{ϵ}_i\right]\cdots \mathrm{E}[X+t]=\mathrm{E}\left[X\right]+t\\ & =& \alpha +\beta x_i+0\cdots {ϵ}_i\stackrel{iid}{\sim }\mathrm{N}(0,{\sigma }^2)\\ & =& \alpha +\beta x_i\\ \mathrm{V}\left[y_i\right]& =& \mathrm{V}[\alpha +\beta x_i+{ϵ}_i]\\ & =& \mathrm{V}\left[{ϵ}_i\right]\cdots \mathrm{V}[X+t]=\mathrm{V}\left[X\right]\\ & =& {\sigma }^2\cdots {ϵ}_i\stackrel{iid}{\sim }\mathrm{N}(0,{\sigma }^2)\\ y_i& \sim & \mathrm{N}(\alpha +\beta x_i,{\sigma }^2)\end{array}$$ $y_i$は$\mathrm{N}(\alpha +\beta x_i,{\sigma }^2)$に従う確率変数である．

$\hat{\beta }$を$\sum _{i=1}^n{c}_i{y}_i$の形で表す

推定量が$\sum _{i=1}^n{c}_i{x}_i(x_i:\mathrm{標}\mathrm{本},c_i:\mathrm{定}\mathrm{数})$の形で表現できるとき，この推定量を線形推定量(linear estimate)という．
(よく知られる線形推定量の例として平均$\overline{x}$があり，$\overline{x}=\sum _{i=1}^n\frac{1}{n}{x}_i$で表現される) $$\begin{array}{rcl}\hat{\beta }& =& \frac{S_{xy}}{S_{xx}}\cdots \hat{\beta }=\frac{S_{xy}}{S_{xx}},S_{xx}=\sum _{i=1}^n{(x_i-\overline{x})}^2,\overline{x}=\frac{1}{n}\sum _{i=1}^n{x}_i\\ & =& \frac{1}{S_{xx}}\sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})\\ & =& \frac{1}{S_{xx}}\sum _{i=1}^n(x_i-\overline{x})y_i\cdots \sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})=S_{xy}=\sum _{i=1}^n(x_i-\overline{x})y_i\\ & =& \sum _{i=1}^n\frac{x_i-\overline{x}}{S_{xx}}{y}_i\\ & =& \sum _{i=1}^n{c}_i{y}_i\cdots c_i=\frac{x_i-\overline{x}}{S_{xx}}\end{array}$$

$\hat{\beta }$の期待値を$\sum _{i=1}^n{c}_i{y}_i$から求めてみる

$$\begin{array}{rcl}\mathrm{E}\left[\sum _{i=1}^n{c}_i{y}_i\right]& =& \mathrm{E}\left[\sum _{i=1}^n\frac{x_i-\overline{x}}{S_{xx}}{y}_i\right]\\ & =& \mathrm{E}\left[\frac{S_{xy}}{S_{xx}}\right]\cdots \mathrm{上}\mathrm{記},\frac{S_{xy}}{S_{xx}}=\sum _{i=1}^n\frac{x_i-\overline{x}}{S_{xx}}{y}_i\\ & =& \mathrm{E}\left[\hat{\beta }\right]\cdots \hat{\beta }=\frac{S_{xy}}{S_{xx}}\\ & =& \beta \cdots \mathrm{E}\left[\hat{\beta }\right]=\beta \end{array}$$

$\hat{\beta }$の分散を$\sum _{i=1}^n{c}_i{y}_i$から求めてみる

$$\begin{array}{rcl}\mathrm{V}\left[\sum _{i=1}^n{c}_i{y}_i\right]& =& \mathrm{V}\left[\sum _{i=1}^n\frac{x_i-\overline{x}}{S_{xx}}{y}_i\right]\\ & =& \sum _{i=1}^n\mathrm{V}\left[\frac{x_i-\overline{x}}{S_{xx}}{y}_i\right]\cdots y_i\mathrm{は}\mathrm{互}\mathrm{い}\mathrm{に}\mathrm{独}\mathrm{立}\mathrm{Cov}[y_i,y_j]=0,\mathrm{互}\mathrm{い}\mathrm{に}\mathrm{独}\mathrm{立}\mathrm{の}\mathrm{場}\mathrm{合}\mathrm{V}[X+Y]=\mathrm{V}\left[X\right]+\mathrm{V}\left[Y\right]\\ & =& \sum _{i=1}^n{\left(\frac{x_i-\overline{x}}{S_{xx}}\right)}^2\mathrm{V}\left[y_i\right]\cdots \mathrm{V}\left[cX\right]=c^2\mathrm{V}\left[X\right]\\ & =& \sum _{i=1}^n{\left(\frac{x_i-\overline{x}}{S_{xx}}\right)}^2{\sigma }^2\\ & =& \frac{{\sigma }^2}{S_{xx}^2}\sum _{i=1}^n{(x_i-\overline{x})}^2\\ & =& \frac{{\sigma }^2}{S_{xx}^2}{S}_{xx}\cdots S_{xx}=\sum _{i=1}^n{(x_i-\overline{x})}^2\\ & =& \frac{{\sigma }^2}{S_{xx}}\cdots \mathrm{V}\left[\frac{S_{xy}}{S_{xx}}\right]=\frac{{\sigma }^2}{S_{xx}}\mathrm{と}\mathrm{同}\mathrm{じ}\mathrm{結}\mathrm{果}\end{array}$$

$\hat{\beta }$の分布

以上のように，$\hat{\beta }$は線形推定量であり，正規分布に従う$y_i$の定数倍の和で表すことができた．よって$\hat{\beta }$は同様に正規分布に従い，その期待値と分散はそれぞれ上記で求めたとおりである $\cdots Z=c_{1}X+c_{2}Y(X\sim \mathrm{N}({\mu }_1,{\sigma }_1^2),Y\sim \mathrm{N}({\mu }_2,{\sigma }_2^2),Z\sim \mathrm{N}(c_1{\mu }_1+c_2{\mu }_2,c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2))$． $$\begin{array}{rcl}\hat{\beta }& \sim & \mathrm{N}(\beta ,\frac{{\sigma }^2}{S_{xx}})\end{array}$$

$\hat{\alpha }$を$\sum _{i=1}^n{c}_i{y}_i$の形で表す

$$\begin{array}{rcl}\hat{\alpha }& =& \overline{y}-\hat{\beta }\overline{x}\cdots \hat{\alpha }=\overline{y}-\hat{\beta }\overline{x}\\ & =& \sum _{i=1}^n\frac{1}{n}{y}_i-\frac{S_{xy}}{S_{xx}}\overline{x}\\ & =& \sum _{i=1}^n\frac{1}{n}{y}_i-\left(\sum _{i=1}^n\frac{x_i-\overline{x}}{S_{xx}}{y}_i\right)\overline{x}\cdots \mathrm{上}\mathrm{記},\frac{S_{xy}}{S_{xx}}=\sum _{i=1}^n\frac{x_i-\overline{x}}{S_{xx}}{y}_i\\ & =& \sum _{i=1}^n\frac{1}{n}{y}_i-\overline{x}\sum _{i=1}^n\frac{x_i-\overline{x}}{S_{xx}}{y}_i\\ & =& \sum _{i=1}^n\frac{1}{n}{y}_i-\sum _{i=1}^n\frac{\overline{x}(x_i-\overline{x})}{S_{xx}}{y}_i\\ & =& \sum _{i=1}^n(\frac{1}{n}-\frac{\overline{x}(x_i-\overline{x})}{S_{xx}})y_i\\ & =& \sum _{i=1}^n{c}_i{y}_i\cdots c_i=\frac{1}{n}-\frac{\overline{x}(x_i-\overline{x})}{S_{xx}}\end{array}$$

$\hat{\alpha }$の期待値を$\sum _{i=1}^n{c}_i{y}_i$から求めてみる

$$\begin{array}{rcl}\mathrm{E}\left[\sum _{i=1}^n{c}_i{y}_i\right]& =& \mathrm{E}\left[\sum _{i=1}^n(\frac{1}{n}-\frac{\overline{x}(x_i-\overline{x})}{S_{xx}})y_i\right]\\ & =& \mathrm{E}\left[\sum _{i=1}^n(\frac{1}{n}{y}_i-\frac{\overline{x}(x_i-\overline{x})}{S_{xx}}{y}_i)\right]\\ & =& \mathrm{E}[\sum _{i=1}^n\frac{1}{n}{y}_i-\sum _{i=1}^n\frac{\overline{x}(x_i-\overline{x})}{S_{xx}}{y}_i]\\ & =& \mathrm{E}[\sum _{i=1}^n\frac{1}{n}{y}_i-\sum _{i=1}^n\frac{\overline{x}(x_i-\overline{x})}{S_{xx}}{y}_i]\\ & =& \mathrm{E}\left[\sum _{i=1}^n\frac{1}{n}{y}_i\right]-\mathrm{E}\left[\sum _{i=1}^n\frac{\overline{x}(x_i-\overline{x})}{S_{xx}}{y}_i\right]\cdots \mathrm{E}[X+Y]=\mathrm{E}\left[X\right]+\mathrm{E}\left[Y\right]\\ & =& \mathrm{E}\left[\frac{1}{n}\sum _{i=1}^n{y}_i\right]-\mathrm{E}\left[\overline{x}\sum _{i=1}^n\frac{(x_i-\overline{x})}{S_{xx}}{y}_i\right]\\ & =& \frac{1}{n}\mathrm{E}\left[\sum _{i=1}^n{y}_i\right]-\overline{x}\mathrm{E}\left[\sum _{i=1}^n\frac{(x_i-\overline{x})}{S_{xx}}{y}_i\right]\cdots \mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]\\ & =& \frac{1}{n}\sum _{i=1}^n\mathrm{E}\left[y_i\right]-\overline{x}\mathrm{E}\left[\frac{S_{xy}}{S_{xx}}\right]\cdots \mathrm{上}\mathrm{記},\frac{S_{xy}}{S_{xx}}=\sum _{i=1}^n\frac{x_i-\overline{x}}{S_{xx}}{y}_i\\ & =& \frac{1}{n}\sum _{i=1}^n(\alpha +\beta x_i)-\overline{x}\mathrm{E}\left[\frac{S_{xy}}{S_{xx}}\right]\cdots \mathrm{E}\left[y_i\right]=\alpha +\beta x_i\\ & =& \frac{1}{n}(\alpha \sum _{i=1}^{n}1+\beta \sum _{i=1}^n{x}_i)-\overline{x}\mathrm{E}\left[\hat{\beta }\right]\cdots \hat{\beta }=\frac{S_{xy}}{S_{xx}}\\ & =& \frac{1}{n}(n\alpha +\beta n\overline{x})-\overline{x}\beta \cdots \mathrm{E}\left[\hat{\beta }\right]=\beta \\ & =& \frac{1}{n}n(\alpha +\beta \overline{x})-\overline{x}\beta \\ & =& (\alpha +\beta \overline{x})-\overline{x}\beta \\ & =& \alpha \cdots \mathrm{E}\left[\hat{\alpha }\right]=\alpha \end{array}$$

$\hat{\alpha }$の分散を$\sum _{i=1}^n{c}_i{y}_i$から求めてみる

$$\begin{array}{rcl}\mathrm{V}\left[\sum _{i=1}^n{c}_i{y}_i\right]& =& \mathrm{V}\left[\sum _{i=1}^n(\frac{1}{n}-\frac{\overline{x}(x_i-\overline{x})}{S_{xx}})y_i\right]\\ & =& \sum _{i=1}^n\mathrm{V}\left[(\frac{1}{n}-\frac{\overline{x}(x_i-\overline{x})}{S_{xx}})y_i\right]\cdots \mathrm{V}\left[\frac{S_{xy}}{S_{xx}}\right]=\frac{{\sigma }^2}{S_{xx}}\mathrm{と}\mathrm{同}\mathrm{じ}\mathrm{結}\mathrm{果}\\ & =& \sum _{i=1}^n{(\frac{1}{n}-\frac{\overline{x}(x_i-\overline{x})}{S_{xx}})}^2\mathrm{V}\left[y_i\right]\cdots \mathrm{V}\left[cX\right]=c^2\mathrm{V}\left[X\right]\\ & =& \sum _{i=1}^n{(\frac{1}{n}-\frac{\overline{x}(x_i-\overline{x})}{S_{xx}})}^2{\sigma }^2\\ & =& {\sigma }^2\sum _{i=1}^n{(\frac{1}{n}-\frac{\overline{x}(x_i-\overline{x})}{S_{xx}})}^2\\ & =& {\sigma }^2\sum _{i=1}^n(\frac{1}{n^2}-2\frac{1}{n}\frac{\overline{x}(x_i-\overline{x})}{S_{xx}}+{\left(\frac{\overline{x}(x_i-\overline{x})}{S_{xx}}\right)}^2)\\ & =& {\sigma }^2(\sum _{i=1}^n\frac{1}{n^2}-\sum _{i=1}^{n}2\frac{1}{n}\frac{\overline{x}(x_i-\overline{x})}{S_{xx}}+\sum _{i=1}^n{\left(\frac{\overline{x}(x_i-\overline{x})}{S_{xx}}\right)}^2)\\ & =& {\sigma }^2(\frac{1}{n^2}\sum _{i=1}^{n}1-\frac{2\overline{x}}{n{S}_{xx}}\sum _{i=1}^n(x_i-\overline{x})+\frac{{\overline{x}}^2}{S_{xx}^2}\sum _{i=1}^n{(x_i-\overline{x})}^2)\\ & =& {\sigma }^2(\frac{1}{n^2}n-\frac{2\overline{x}}{n{S}_{xx}}\cdot 0+\frac{{\overline{x}}^2}{S_{xx}^2}{S}_{xx})\\ & =& {\sigma }^2(\frac{1}{n^2}\cdot n-\frac{2\overline{x}}{n{S}_{xx}}\cdot 0+\frac{{\overline{x}}^2}{S_{xx}^2}\cdot S_{xx})\\ & =& {\sigma }^2(\frac{1}{n}+\frac{{\overline{x}}^2}{S_{xx}})\cdots \mathrm{V}[\overline{y}-\hat{\beta }\overline{x}]=(\frac{1}{n}+\frac{{\overline{x}}^2}{S_{xx}}){\sigma }^2\mathrm{と}\mathrm{同}\mathrm{じ}\mathrm{結}\mathrm{果}\end{array}$$

$\hat{\alpha }$の分布

以上のように，$\hat{\alpha }$は線形推定量であり，正規分布に従う$y_i$の定数倍の和で表すことができた．よって$\hat{\alpha }$は同様に正規分布に従い，その期待値と分散はそれぞれ上記で求めたとおりである $\cdots Z=c_{1}X+c_{2}Y(X\sim \mathrm{N}({\mu }_1,{\sigma }_1^2),Y\sim \mathrm{N}({\mu }_2,{\sigma }_2^2),Z\sim \mathrm{N}(c_1{\mu }_1+c_2{\mu }_2,c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2))$． $$\begin{array}{rcl}\hat{\alpha }& \sim & \mathrm{N}(\alpha ,{\sigma }^2(\frac{1}{n}+\frac{{\overline{x}}^2}{S_{xx}}))\end{array}$$

確率変数の変数変換 Z=c1X+c2Y / 正規分布の定数倍同士の和

確率変数の変数変換 $Z=c_{1}X+c_{2}Y$ / 正規分布の定数倍同士の和

確率変数の変数変換 $Z=c_{1}X+c_{2}Y$

$$\begin{array}{rcl}{p}_{c_{1}X+c_{2}Y}(z)& =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (z-(c_{1}x+c_{2}y))f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y\\ & & \cdots {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)f(x)\mathrm{d}x=f(0)\\ & & \cdots X=x,Y=y\mathrm{の}\mathrm{同}\mathrm{時}\mathrm{確}\mathrm{率}f(x)g(x)\mathrm{の}\mathrm{う}\mathrm{ち}z-(c_{1}x+c_{2}y)=0\mathrm{を}\mathrm{満}\mathrm{た}\mathrm{す}\mathrm{も}\mathrm{の}\mathrm{だ}\mathrm{け}\mathrm{を}\mathrm{足}\mathrm{し}\mathrm{合}\mathrm{わ}\mathrm{せ}\mathrm{る}.\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (z-(c_{1}x+c_{2}y))f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y\\ & & \cdots \delta (u(x))=\sum _{\alpha \in u^{-1}(0)}\frac{1}{|u^{\prime }(\alpha )|}\delta (x-\alpha )\\ & & \cdots u(x)=z-(c_{1}x+c_{2}y)\\ & & \cdots u(x=\alpha )=0,\alpha =\frac{z-c_{2}y}{c_1}\\ & & \cdots u^{\prime }=\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{\mathrm{d}}{\mathrm{d}x}(z-(c_{1}x+c_{2}y))=-c_1\\ & & \cdots \delta (z-(c_{1}x+c_{2}y))=\frac{1}{|u^{\prime }(\alpha )|}\delta (x-\alpha )=\frac{1}{|-c_1|}\delta (x-\frac{z-c_{2}y}{c_1})=\frac{1}{\left|c_1\right|}\delta (x-\frac{z-c_{2}y}{c_1})\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{\left|c_1\right|}\delta (x-\frac{z-c_{2}y}{c_1})f_X\left(x\right)g_Y(y)\mathrm{d}y\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{\left|c_1\right|}\delta (\frac{z-c_{2}y}{c_1}-\frac{z-c_{2}y}{c_1})f_X\left(\frac{z-c_{2}y}{c_1}\right)g_Y(y)\mathrm{d}y\cdots z=c_{1}x+c_{2}y,x=\frac{z-c_{2}y}{c_1}\\ & =& \frac{1}{\left|c_1\right|}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (0)f_X\left(\frac{z-c_{2}y}{c_1}\right)g_Y(y)\mathrm{d}y\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{\left|c_1\right|}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_X\left(\frac{z-c_{2}y}{c_1}\right)g_Y(y)\mathrm{d}y\end{array}$$

正規分布の定数倍同士の和の例

$$\begin{array}{rcl}{f}_X(x)& =& \frac{1}{\sqrt{2\pi {\sigma }_1^2}}{\mathrm{e}}^{\frac{-(x-{\mu }_1{)}^2}{2{\sigma }_1^2}}\cdots \mathrm{N}({\mu }_1,{\sigma }_1^2)(\mathrm{正}\mathrm{規}\mathrm{分}\mathrm{布})\\ g_Y(y)& =& \frac{1}{\sqrt{2\pi {\sigma }_2^2}}{\mathrm{e}}^{\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\cdots \mathrm{N}({\mu }_2,{\sigma }_2^2)(\mathrm{正}\mathrm{規}\mathrm{分}\mathrm{布})\\ p_{c_{1}X+c_{2}Y}(z)& =& \frac{1}{\left|c_1\right|}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_X\left(\frac{z-c_{2}y}{c_1}\right)g_Y(y)\mathrm{d}y\\ & =& \frac{1}{\left|c_1\right|}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{\sqrt{2\pi {\sigma }_1^2}}{\mathrm{e}}^{\frac{-{\{\left(\frac{z-c_{2}y}{c_1}\right)-{\mu }_1\}}^2}{2{\sigma }_1^2}}\frac{1}{\sqrt{2\pi {\sigma }_2^2}}{\mathrm{e}}^{\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\mathrm{d}y\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{\sqrt{2\pi {\sigma }_1^2}}\frac{1}{\sqrt{2\pi {\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\frac{-{\{\left(\frac{z-c_{2}y}{c_1}\right)-{\mu }_1\}}^2}{2{\sigma }_1^2}+\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\mathrm{d}y\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\frac{-{\{\left(\frac{z-c_{2}y}{c_1}\right)-{\mu }_1\}}^2}{2{\sigma }_1^2}+\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\mathrm{d}y\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{f_1(c_1,c_2,y,z,{\mu }_1,{\mu }_2,{\sigma }_1,{\sigma }_2)}\mathrm{d}y\cdots f_1\mathrm{は}y\mathrm{が}\mathrm{引}\mathrm{数}\mathrm{に}\mathrm{あ}\mathrm{る}．\end{array}$$

---

$$\begin{array}{rcl}{f}_1(c_1,c_2,y,z,{\mu }_1,{\mu }_2,{\sigma }_1,{\sigma }_2)& =& \frac{-{\{\left(\frac{z-c_{2}y}{c_1}\right)-{\mu }_1\}}^2}{2{\sigma }_1^2}+\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}\\ & =& -\frac{1}{2}\{\frac{{(\frac{1}{c_1}z-\frac{c_2}{c_1}y-{\mu }_1)}^2}{{\sigma }_1^2}+\frac{(y-{\mu }_2{)}^2}{{\sigma }_2^2}\}\\ & =& -\frac{1}{2}\left\{\frac{{\sigma }_2^2{(\frac{1}{c_1}z-\frac{c_2}{c_1}y-{\mu }_1)}^2+{\sigma }_1^2(y-{\mu }_2{)}^2}{{\sigma }_1^2{\sigma }_2^2}\right\}\\ & =& -\frac{1}{2{\sigma }_1^2{\sigma }_2^2}\{{\sigma }_2^2{(\frac{1}{c_1}z-\frac{c_2}{c_1}y-{\mu }_1)}^2+{\sigma }_1^2(y-{\mu }_2{)}^2\}\\ & =& -\frac{1}{2{\sigma }_1^2{\sigma }_2^2}[{\sigma }_2^2\{{\left(\frac{1}{c_1}z\right)}^2-2\left(\frac{1}{c_1}z\right)\left(\frac{c_2}{c_1}y\right)-2\left(\frac{1}{c_1}z\right){\mu }_1+{\left(\frac{c_2}{c_1}y\right)}^2+2\left(\frac{c_2}{c_1}y\right){\mu }_1+{\mu }_1^2\}+{\sigma }_1^2(y^2-2y{\mu }_2+{\mu }_2^2)]\\ & =& -\frac{1}{2{\sigma }_1^2{\sigma }_2^2}\{\frac{1}{c_1^2}{\sigma }_2^2{z}^2-2\frac{c_2}{c_1^2}{\sigma }_2^{2}zy-2\frac{1}{c_1}{\sigma }_2^{2}z{\mu }_1+\frac{c_2^2}{c_1^2}{\sigma }_2^2{y}^2+2\frac{c_2}{c_1}{\sigma }_2^{2}y{\mu }_1+{\sigma }_2^2{\mu }_1^2+{\sigma }_1^2{y}^2-2{\sigma }_1^{2}y{\mu }_2+{\sigma }_1^2{\mu }_2^2\}\\ & =& -\frac{1}{2{\sigma }_1^2{\sigma }_2^2}\frac{1}{c_1^2}\{{\sigma }_2^2{z}^2-2c_2{\sigma }_2^{2}zy-2c_1{\sigma }_2^{2}z{\mu }_1+c_2^2{\sigma }_2^2{y}^2+2c_1{c}_2{\sigma }_2^{2}y{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{y}^2-2c_1^2{\sigma }_1^{2}y{\mu }_2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& -\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)y^2+(-2c_2{\sigma }_2^{2}z+2c_1{c}_2{\sigma }_2^2{\mu }_1-2c_1^2{\sigma }_1^2{\mu }_2)y+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& -\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)y^2-2(c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2)y+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& -\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)(y^2-2\frac{c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}y)+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& \\ & =& -\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)(y^2-2\frac{c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}y)+A-A+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & & \cdots \mathrm{平}\mathrm{方}\mathrm{完}\mathrm{成}\mathrm{さ}\mathrm{せ}\mathrm{る}\mathrm{た}\mathrm{め}\mathrm{の}\mathrm{項}A\mathrm{を}\mathrm{考}\mathrm{え}\mathrm{る}\\ & =& -\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)(y^2-2\frac{c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}y+\frac{1}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}A)-A+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& -\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}[(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)\{y^2-2\frac{c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}y+\frac{1}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}\frac{{(c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2)}^2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}\}-\frac{{(c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2)}^2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2]\\ & & \cdots A=\frac{{(c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2)}^2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}\\ & =& -\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2){(y^2-\frac{c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2})}^2-\frac{{(c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2)}^2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& -\frac{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}{2c_1^2{\sigma }_1^2{\sigma }_2^2}{(y^2-\frac{c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2})}^2-\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{-\frac{{(c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2)}^2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& -f_2(c_1,c_2,{\sigma }_1^2,{\sigma }_2^2){\{y-f_3(c_1,c_2,z,{\mu }_1,{\mu }_2,{\sigma }_1^2,{\sigma }_2^2)\}}^2-f_4(c_1,c_2,z,{\mu }_1,{\mu }_2,{\sigma }_1^2,{\sigma }_2^2)\cdots f_2,f_3,f_4\mathrm{は}y\mathrm{を}\mathrm{引}\mathrm{数}\mathrm{と}\mathrm{し}\mathrm{な}\mathrm{い}.\\ & =& -f_2{(y-f_3)}^2-f_4\cdots y\mathrm{が}\mathrm{平}\mathrm{方}\mathrm{完}\mathrm{成}\mathrm{の}\mathrm{中}\mathrm{の}\mathrm{一}\mathrm{つ}\mathrm{だ}\mathrm{け}\mathrm{に}\mathrm{な}\mathrm{っ}\mathrm{て}\mathrm{い}\mathrm{る}.\\ \\ f_2(c_1,c_2,{\sigma }_1^2,{\sigma }_2^2)& =& \frac{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\\ \\ f_3(c_1,c_2,z,{\mu }_1,{\mu }_2,{\sigma }_1^2,{\sigma }_2^2)& =& \frac{c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}\\ \\ f_4(c_1,c_2,z,{\mu }_1,{\mu }_2,{\sigma }_1^2,{\sigma }_2^2)& =& \frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{-\frac{{(c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2)}^2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& \frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\left\{\frac{-{(c_2{\sigma }_2^2(z-c_1{\mu }_1)+c_1^2{\sigma }_1^2{\mu }_2)}^2+(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)({\sigma }_2^2{(z-c_1{\mu }_1)}^2+c_1^2{\sigma }_1^2{\mu }_2^2)}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}\right\}\\ & =& \frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\frac{-{(c_2{\sigma }_2^{2}B+c_1^2{\sigma }_1^2{\mu }_2)}^2+(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)({\sigma }_2^2{B}^2+c_1^2{\sigma }_1^2{\mu }_2^2)}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}\cdots B=z-c_1{\mu }_1\\ & =& \frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}\{-{(c_2{\sigma }_2^{2}B+c_1^2{\sigma }_1^2{\mu }_2)}^2+(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)({\sigma }_2^2{B}^2+c_1^2{\sigma }_1^2{\mu }_2^2)\}\\ & =& \frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}\left(-c_2^2{\sigma }_2^4{B}^2-2c_1^2{c}_2{\sigma }_1^2{\sigma }_2^2{\mu }_{2}B-c_1^4{\sigma }_1^4{\mu }_2^2+c_1^2{\sigma }_1^2{\sigma }_2^2{B}^2+c_2^2{\sigma }_2^4{B}^2+c_1^4{\sigma }_1^4{\mu }_2^2+c_1^2{c}_2^2{\sigma }_1^2{\sigma }_2^2{\mu }_2^2\right)\\ & =& \frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}{c}_1^2{\sigma }_1^2{\sigma }_2^2\{-2c_2{\mu }_{2}B+B^2+c_2^2{\mu }_2^2\}\\ & =& \frac{1}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}{(B-c_2{\mu }_2)}^2\\ & =& \frac{{(z-c_1{\mu }_1-c_2{\mu }_2)}^2}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}\cdots B=z-c_1{\mu }_1\\ & =& \frac{{(z-(c_1{\mu }_1+c_2{\mu }_2))}^2}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}\end{array}$$

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$$\begin{array}{rcl}{p}_{c_{1}X+c_{2}Y}(z)& =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{f_1(c_1,c_2,y,z,{\mu }_1,{\mu }_2,{\sigma }_1,{\sigma }_2)}\mathrm{d}y\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-f_2{(y-f_3)}^2-f_4}\mathrm{d}y\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-f_2{(y-f_3)}^2}{\mathrm{e}}^{-f_4}\mathrm{d}y\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-f_2{(y-f_3)}^2}\mathrm{d}y\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-u^2}\frac{1}{\sqrt{f_2}}\mathrm{d}u\\ & & \cdots u=\sqrt{f_2}(y-f_3),\frac{\mathrm{d}u}{\mathrm{d}y}=\sqrt{f_2},\mathrm{d}y=\frac{1}{\sqrt{f_2}}\mathrm{d}u\\ & & \cdots y:-\mathrm{\infty }\to \mathrm{\infty },u:-\mathrm{\infty }\to \mathrm{\infty }\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}\frac{1}{\sqrt{f_2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-u^2}\mathrm{d}u\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}\frac{1}{\sqrt{f_2}}\sqrt{\pi }\cdots {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-u^2}\mathrm{d}u=\sqrt{\pi }\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}\frac{\sqrt{\pi }}{\sqrt{f_2}}\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}\sqrt{\frac{\pi }{f_2}}{\mathrm{e}}^{-f_4}\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}\sqrt{\frac{\pi }{\frac{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}{2c_1^2{\sigma }_1^2{\sigma }_2^2}}}{\mathrm{e}}^{-\frac{{(z-(c_1{\mu }_1+c_2{\mu }_2))}^2}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}\sqrt{\frac{2\pi c_1^2{\sigma }_1^2{\sigma }_2^2}{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}{\mathrm{e}}^{-\frac{{(z-(c_1{\mu }_1+c_2{\mu }_2))}^2}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}\\ & =& \frac{1}{\left|c_1\right|}\frac{\sqrt{c_1^2}}{\sqrt{2\pi (c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}{\mathrm{e}}^{-\frac{{(z-(c_1{\mu }_1+c_2{\mu }_2))}^2}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}\\ & =& \frac{1}{\left|c_1\right|}\frac{\left|c_1\right|}{\sqrt{2\pi (c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}{\mathrm{e}}^{-\frac{{(z-(c_1{\mu }_1+c_2{\mu }_2))}^2}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}\cdots \sqrt{A^2}=\{\begin{array}{}A& (A\ge 0)\\ -A& (A<0)\end{array}=\left|A\right|\\ & =& \frac{1}{\sqrt{2\pi (c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}{\mathrm{e}}^{-\frac{{(z-(c_1{\mu }_1+c_2{\mu }_2))}^2}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}\\ & \sim & \mathrm{N}(c_1{\mu }_1+c_2{\mu }_2,c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)\end{array}$$

確率変数の変数変換 Z=X+Y その2 / 正規分布の再生性

確率変数の変数変換 Z=X+Y その2 / 正規分布の再生性

Z=X+Y 正規分布同士の例

$$\begin{array}{rcl}{f}_X(x)& =& \frac{1}{\sqrt{2\pi {\sigma }_1^2}}{\mathrm{e}}^{\frac{-(x-{\mu }_1{)}^2}{2{\sigma }_1^2}}\cdots \mathrm{N}({\mu }_1,{\sigma }_1^2)(\mathrm{正}\mathrm{規}\mathrm{分}\mathrm{布})\\ g_Y(x)& =& \frac{1}{\sqrt{2\pi {\sigma }_2^2}}{\mathrm{e}}^{\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\cdots \mathrm{N}({\mu }_2,{\sigma }_2^2)(\mathrm{正}\mathrm{規}\mathrm{分}\mathrm{布})\\ p_{X+Y}(z)& =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (z-(x+y))f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (z-((z-y)+y))f_X(z-y)g_Y(y)\mathrm{d}y\cdots z=x+y,x=z-y\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (0)f_X(z-y)g_Y(y)\mathrm{d}y\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_X(z-y)g_Y(y)\mathrm{d}y\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{\sqrt{2\pi {\sigma }_1^2}}{\mathrm{e}}^{\frac{-(z-y-{\mu }_1{)}^2}{2{\sigma }_1^2}}\frac{1}{\sqrt{2\pi {\sigma }_2^2}}{\mathrm{e}}^{\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\mathrm{d}y\\ & =& \frac{1}{\sqrt{2\pi {\sigma }_1^2}}\frac{1}{\sqrt{2\pi {\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\frac{-(z-y-{\mu }_1{)}^2}{2{\sigma }_1^2}+\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\mathrm{d}y\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\frac{-(z-y-{\mu }_1{)}^2}{2{\sigma }_1^2}+\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\mathrm{d}y\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{f_1(y,z,{\mu }_1,{\mu }_2,{\sigma }_1,{\sigma }_2)}\mathrm{d}y\cdots f_1\mathrm{は}y\mathrm{が}\mathrm{引}\mathrm{数}\mathrm{に}\mathrm{あ}\mathrm{る}．\end{array}$$

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$$\begin{array}{rcl}{p}_{X+Y}(z)& =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{f_1(y,z,{\mu }_1,{\mu }_2,{\sigma }_1,{\sigma }_2)}\mathrm{d}y\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-f_2{(y-f_3)}^2-f_4}\mathrm{d}y\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-f_2{(y-f_3)}^2}{\mathrm{e}}^{-f_4}\mathrm{d}y\cdots A^{B+C}=A^B{A}^C\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-f_2{(y-f_3)}^2}\mathrm{d}y\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-f_2{(y-f_3)}^2}\mathrm{d}y\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-u^2}\left(\frac{1}{\sqrt{f_2}}\right)\mathrm{d}u\\ & & \cdots u=\sqrt{f_2}(y-f_3),\frac{\mathrm{d}u}{\mathrm{d}y}=\sqrt{f_2},\mathrm{d}y=\frac{1}{\sqrt{f_2}}\mathrm{d}u,\\ & & \cdots y:-\mathrm{\infty }\to \mathrm{\infty },u:-\mathrm{\infty }\to \mathrm{\infty }\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}\frac{1}{\sqrt{f_2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-u^2}\mathrm{d}u\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}\frac{1}{\sqrt{f_2}}{\mathrm{e}}^{-f_4}\sqrt{\pi }\cdots {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-u^2}\mathrm{d}u=\sqrt{\pi }\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}\frac{1}{\sqrt{\frac{{\sigma }_1^2+{\sigma }_2^2}{2{\sigma }_1^2{\sigma }_2^2}}}{\mathrm{e}}^{-\frac{{\{z-({\mu }_1+{\mu }_2)\}}^2}{2({\sigma }_1^2+{\sigma }_2^2)}}\sqrt{\pi }\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}\sqrt{\frac{2\pi {\sigma }_1^2{\sigma }_2^2}{{\sigma }_1^2+{\sigma }_2^2}}{\mathrm{e}}^{-\frac{{\{z-({\mu }_1+{\mu }_2)\}}^2}{2({\sigma }_1^2+{\sigma }_2^2)}}\\ & =& \frac{1}{\sqrt{2\pi ({\sigma }_1^2+{\sigma }_2^2)}}{\mathrm{e}}^{-\frac{{\{z-({\mu }_1+{\mu }_2)\}}^2}{2({\sigma }_1^2+{\sigma }_2^2)}}\cdots \mathrm{N}({\mu }_1+{\mu }_2,{\sigma }_1^2+{\sigma }_2^2)(\mathrm{再}\mathrm{生}\mathrm{性})\end{array}$$ 再生性は，同じ確率分布族に含まれる確率分布$F_1$,$F_2$に対して $X_1\sim F_1$,$X_2\sim F_2$とする互いに独立な確率変数に対して， $X_1+X_2$がやはり同一の確率分布族に含まれる性質．

単回帰モデルの最尤推定量の期待値,分散,分布

単回帰モデルの最尤推定量の期待値,分散,分布

単回帰モデル

$$\begin{array}{rcl}{y}_i& =& \alpha +\beta x_i+{ϵ}_i(i=1,\cdots ,n)\\ & & {ϵ}_i\stackrel{iid}{\sim }N(0,{\sigma }^2)\cdots \mathrm{独}\mathrm{立}\mathrm{同}\mathrm{一}\mathrm{分}\mathrm{布}(independentandidenticallydistributed;IID,i.i.d.,iid)\end{array}$$ $\alpha ,\beta ,{\sigma }^2$の推定量を$\hat{\alpha },\hat{\beta },{\hat{\sigma }}^2$とし，$\hat{\alpha },\hat{\beta },{\hat{\sigma }}^2$の最尤推定量(maximum likelihood estimator)を${\hat{\alpha }}_{ML},{\hat{\beta }}_{ML},{\hat{\sigma }}_{ML}^2$とする．

${\hat{\beta }}_{ML}$の期待値

$$\begin{array}{rcl}\mathrm{E}\left[{\hat{\beta }}_{ML}\right]& =& \mathrm{E}\left[\hat{\beta }\right]\cdots {\hat{\beta }}_{ML}=\hat{\beta }\\ & =& \beta \cdots \mathrm{E}\left[\hat{\beta }\right]=\beta \\ & & \cdots \mathrm{よ}\mathrm{っ}\mathrm{て}{\hat{\beta }}_{ML}^2\mathrm{は}{\beta }^2\mathrm{の}\mathrm{不}\mathrm{偏}\mathrm{推}\mathrm{定}\mathrm{量}\mathrm{で}\mathrm{あ}\mathrm{る}.\end{array}$$

${\hat{\alpha }}_{ML}$の期待値

$$\begin{array}{rcl}\mathrm{E}\left[{\hat{\alpha }}_{ML}\right]& =& \mathrm{E}\left[\hat{\alpha }\right]\cdots {\hat{\alpha }}_{ML}=\hat{\alpha }\\ & =& \alpha \cdots \mathrm{E}\left[\hat{\alpha }\right]=\alpha \\ & & \cdots \mathrm{よ}\mathrm{っ}\mathrm{て}{\hat{\alpha }}_{ML}^2\mathrm{は}{\alpha }^2\mathrm{の}\mathrm{不}\mathrm{偏}\mathrm{推}\mathrm{定}\mathrm{量}\mathrm{で}\mathrm{あ}\mathrm{る}.\end{array}$$

${\hat{\beta }}_{ML}$の分散

$$\begin{array}{rcl}\mathrm{V}\left[{\hat{\beta }}_{ML}\right]& =& \mathrm{V}\left[\hat{\beta }\right]\\ & =& \frac{1}{S_{xx}}{\sigma }^2\cdots \mathrm{V}\left[\hat{\beta }\right]=\frac{1}{S_{xx}}{\sigma }^2\\ & & \cdots \overline{x}=\frac{1}{n}\sum _{i=0}^n{x}_i,S_{xx}=\sum _{i=0}^n{(x_i-\overline{x})}^2\end{array}$$

${\hat{\alpha }}_{ML}$の分散

$$\begin{array}{rcl}\mathrm{V}\left[{\hat{\alpha }}_{ML}\right]& =& \mathrm{V}\left[\hat{\alpha }\right]\\ & =& (\frac{1}{n}+\frac{{\overline{x}}^2}{S_{xx}}){\sigma }^2\cdots \mathrm{V}\left[\hat{\alpha }\right]=(\frac{1}{n}+\frac{{\overline{x}}^2}{S_{xx}}){\sigma }^2\end{array}$$

${\hat{\alpha }}_{ML},{\hat{\beta }}_{ML}$の分布

$$\begin{array}{rclrc}{\hat{\beta }}_{ML}& =& \hat{\beta }& \sim & \mathrm{N}(\beta ,\frac{1}{S_{xx}}{\sigma }^2)\cdots \hat{\beta }\sim \mathrm{N}(\beta ,\frac{1}{S_{xx}}{\sigma }^2)\\ {\hat{\alpha }}_{ML}& =& \hat{\alpha }& \sim & \mathrm{N}(\alpha ,(\frac{1}{n}+\frac{{\overline{x}}^2}{S_{xx}}){\sigma }^2)\cdots \hat{\alpha }\sim \mathrm{N}(\alpha ,(\frac{1}{n}+\frac{{\overline{x}}^2}{S_{xx}}){\sigma }^2)\end{array}$$

${\hat{\sigma }}_{ML}^2$の期待値

$$\begin{array}{rcl}\mathrm{E}\left[{\hat{\sigma }}_{ML}^2\right]& =& \mathrm{E}\left[\frac{n-2}{n}{s}^2\right]\cdots {\hat{\sigma }}_{ML}^2=\frac{n-2}{n}{s}^2,s^2=\frac{1}{(n-2)}\sum _{i=1}^n{e}_i^2,\sum _{i=1}^n{e}_i^2=\sum _{i=1}^n{(y_i-\widehat{y_i})}^2\\ & =& \frac{n-2}{n}\mathrm{E}\left[s^2\right]\\ & =& \frac{n-2}{n}{\sigma }^2\cdots \mathrm{E}\left[s^2\right]={\sigma }^2\\ & <& {\sigma }^2\cdots \mathrm{よ}\mathrm{っ}\mathrm{て}{\hat{\sigma }}_{ML}^2\mathrm{は}{\sigma }^2\mathrm{の}\mathrm{不}\mathrm{偏}\mathrm{推}\mathrm{定}\mathrm{量}\mathrm{で}\mathrm{は}\mathrm{な}\mathrm{い}.\end{array}$$

確率変数の標準化

確率変数の標準化

期待値(平均)が$\mu$, 分散が${\sigma }^2$の確率変数$X$

$$\begin{array}{rcl}\mathrm{E}\left[X\right]& =& \mu \\ \mathrm{V}\left[X\right]& =& {\sigma }^2\end{array}$$

確率変数の変換$Z=\frac{X-\mu }{\sigma }$

$$\begin{array}{r}\\ Z& =& \frac{X-\mu }{\sigma }\end{array}$$

変換後の確率変数$Z$の期待値(平均)と分散

$$\begin{array}{r}\\ \mathrm{E}\left[Z\right]& =& \mathrm{E}\left[\frac{X-\mu }{\sigma }\right]\\ & =& \frac{1}{\sigma }\mathrm{E}[X-\mu ]\cdots \mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]\\ & =& \frac{1}{\sigma }(\mathrm{E}\left[X\right]-\mu )\cdots \mathrm{E}[X\pm t]=\mathrm{E}\left[X\right]\pm t\\ & =& \frac{1}{\sigma }(\mu -\mu )\cdots \mathrm{E}\left[X\right]=\mu \\ & =& \frac{1}{\sigma }\left(0\right)\\ & =& 0\\ \mathrm{V}\left[Z\right]& =& \mathrm{V}\left[\frac{X-\mu }{\sigma }\right]\\ & =& \frac{1}{{\sigma }^2}\mathrm{V}[X-\mu ]\cdots \mathrm{V}\left[cX\right]=c^2\mathrm{V}\left[X\right]\\ & =& \frac{1}{{\sigma }^2}\mathrm{V}\left[X\right]\cdots \mathrm{V}[X\pm t]=\mathrm{V}\left[X\right]\\ & =& \frac{1}{{\sigma }^2}{\sigma }^2\cdots \mathrm{V}\left[X\right]={\sigma }^2\\ & =& 1\end{array}$$ $X$の分布によらず，$X$の期待値(平均)と分散が$\mu$と${\sigma }^2$であることから$Z$の期待値(平均)と分散が$0$, $1$と標準化される．

単回帰モデルの最尤推定

単回帰モデルの最尤推定

単回帰モデル

$$\begin{array}{rcl}{y}_i& =& \alpha +\beta x_i+{ϵ}_i(i=1,\cdots ,n)\\ & & {ϵ}_i\stackrel{iid}{\sim }N(0,{\sigma }^2)\cdots \mathrm{独}\mathrm{立}\mathrm{同}\mathrm{一}\mathrm{分}\mathrm{布}(independentandidenticallydistributed;IID,i.i.d.,iid)\end{array}$$

対数尤度凾数

対数尤度凾数は以下のようになる． $$\begin{array}{rcl}f(y_1,\cdots ,y_n;\alpha ,\beta ,{\sigma }^2)& =& \prod _{i=1}^n\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{1}{2{\sigma }^2}{(y_i-\alpha -\beta x_i)}^2}\\ & =& {\left(2\pi \right)}^{-\frac{n}{2}}{\left({\sigma }^2\right)}^{-\frac{n}{2}}{e}^{-\frac{1}{2{\sigma }^2}\sum _{i=1}^n{(y_i-\alpha -\beta x_i)}^2}\\ l(\alpha ,\beta ,{\sigma }^2;y_1,\cdots ,y_n)& =& \log \left\{{\left(2\pi \right)}^{-\frac{n}{2}}{\left({\sigma }^2\right)}^{-\frac{n}{2}}{e}^{-\frac{1}{2{\sigma }^2}\sum _{i=1}^n{(y_i-\alpha -\beta x_i)}^2}\right\}\\ & =& \log \left\{{\left(2\pi \right)}^{-\frac{n}{2}}\right\}+\log \left\{{\left({\sigma }^2\right)}^{-\frac{n}{2}}\right\}+\log \left\{e^{-\frac{1}{2{\sigma }^2}\sum _{i=1}^n{(y_i-\alpha -\beta x_i)}^2}\right\}\\ & =& -\frac{n}{2}\log \left(2\pi \right)-\frac{n}{2}\log \left({\sigma }^2\right)-\frac{1}{2{\sigma }^2}\sum _{i=1}^n{(y_i-\alpha -\beta x_i)}^2\end{array}$$

スコア凾数

スコア凾数は以下のようになる． $$\begin{array}{rcl}\frac{\partial l}{\partial \alpha }& =& \frac{\partial l}{\partial \alpha }\{-\frac{1}{2{\sigma }^2}\sum _{i=1}^n{(y_i-\alpha -\beta x_i)}^2\}\\ & =& -\frac{1}{2{\sigma }^2}\sum _{i=1}^n\frac{\partial l}{\partial \alpha }{(y_i-\alpha -\beta x_i)}^2\\ & =& -\frac{1}{2{\sigma }^2}\sum _{i=1}^n(y_i-\alpha -\beta x_i)(-1)\\ & =& -\frac{-1}{2{\sigma }^2}\sum _{i=1}^n(y_i-\alpha -\beta x_i)\\ & =& \frac{1}{2{\sigma }^2}(\sum _{i=1}^n{y}_i-\alpha \sum _{i=1}^{n}1-\beta \sum _{i=1}^n{x}_i)\\ & =& \frac{1}{2{\sigma }^2}(n\overline{y}-n\alpha -n\beta \overline{x})\cdots \overline{x}=\frac{1}{n}\sum _{i=1}^n{x}_i,\overline{y}=\frac{1}{n}\sum _{i=1}^n{y}_i\\ & =& \frac{n}{2{\sigma }^2}(\overline{y}-\alpha -\beta \overline{x})\end{array}$$ $$\begin{array}{rcl}\frac{\partial l}{\partial \beta }& =& \frac{\partial l}{\partial \beta }\{-\frac{1}{2{\sigma }^2}\sum _{i=1}^n{(y_i-\alpha -\beta x_i)}^2\}\\ & =& -\frac{1}{2{\sigma }^2}\sum _{i=1}^n\frac{\partial l}{\partial \beta }{(y_i-\alpha -\beta x_i)}^2\\ & =& -\frac{1}{2{\sigma }^2}\sum _{i=1}^{n}2(y_i-\alpha -\beta x_i)(-x_i)\\ & =& -\frac{-2}{2{\sigma }^2}\sum _{i=1}^n(x_i{y}_i-x_i\alpha -\beta x_i^2)\\ & =& \frac{1}{{\sigma }^2}(\sum _{i=1}^n{x}_i{y}_i-\sum _{i=1}^n{x}_i\alpha -\sum _{i=1}^n\beta x_i^2)\\ & =& \frac{1}{{\sigma }^2}(\sum _{i=1}^n{x}_i{y}_i-n\overline{x}\alpha -\beta \sum _{i=1}^n{x}_i^2)\cdots \overline{x}=\frac{1}{n}\sum _{i=1}^n{x}_i\end{array}$$ $$\begin{array}{rcl}\frac{\partial l}{\partial {\sigma }^2}& =& \frac{\partial l}{\partial {\sigma }^2}\{-\frac{n}{2}\log \left({\sigma }^2\right)-\frac{1}{2{\sigma }^2}\sum _{i=1}^n{(y_i-\alpha -\beta x_i)}^2\}\\ & =& -\frac{n}{2}\frac{\partial l}{\partial {\sigma }^2}\log \left({\sigma }^2\right)-\frac{1}{2}\left\{\sum _{i=1}^n{(y_i-\alpha -\beta x_i)}^2\right\}\frac{\partial l}{\partial {\sigma }^2}\frac{1}{{\sigma }^2}\\ & =& -\frac{n}{2}\frac{\partial l}{\partial {\sigma }^2}\log \left({\sigma }^2\right)-\frac{1}{2}\left\{\sum _{i=1}^n{(y_i-\alpha -\beta x_i)}^2\right\}\frac{\partial l}{\partial u}\frac{1}{u}\cdots u={\sigma }^2\\ & =& -\frac{n}{2}\frac{\partial l}{\partial {\sigma }^2}\log \left({\sigma }^2\right)-\frac{1}{2}\left\{\sum _{i=1}^n{(y_i-\alpha -\beta x_i)}^2\right\}\left(\frac{-1}{u^2}\right)\\ & =& -\frac{n}{2}\frac{1}{{\sigma }^2}-\frac{1}{2}\left\{\sum _{i=1}^n{(y_i-\alpha -\beta x_i)}^2\right\}\left(\frac{-1}{{\sigma }^4}\right)\cdots u={\sigma }^2\\ & =& -\frac{1}{2{\sigma }^2}\{n-\frac{1}{{\sigma }^2}\sum _{i=1}^n{(y_i-\alpha -\beta x_i)}^2\}\end{array}$$

スコア凾数を連立させる

$$\begin{array}{r}\{\begin{array}{rcl}0& =& \frac{\partial l}{\partial \alpha }\\ 0& =& \frac{\partial l}{\partial \beta }\\ 0& =& \frac{\partial l}{\partial {\sigma }^2}\end{array}\end{array}$$ $$\begin{array}{r}\{\begin{array}{rcl}0& =& \frac{n}{2{\sigma }^2}(\overline{y}-\alpha -\beta \overline{x})\\ 0& =& \frac{1}{{\sigma }^2}(\sum _{i=1}^n{x}_i{y}_i-n\overline{x}\alpha -\beta \sum _{i=1}^n{x}_i^2)\\ 0& =& -\frac{1}{2{\sigma }^2}\{n-\frac{1}{{\sigma }^2}\sum _{i=1}^n{(y_i-\alpha -\beta x_i)}^2\}\end{array}\end{array}$$ $\alpha ,\beta ,{\sigma }^2$の推定量を$\hat{\alpha },\hat{\beta },{\hat{\sigma }}^2$とし，$\hat{\alpha },\hat{\beta },{\hat{\sigma }}^2$の最尤推定量(maximum likelihood estimator)を${\hat{\alpha }}_{ML},{\hat{\beta }}_{ML},{\hat{\sigma }}_{ML}^2$とする． $$\begin{array}{r}\{\begin{array}{rcl}0& =& \overline{y}-{\hat{\alpha }}_{ML}-{\hat{\beta }}_{ML}\overline{x}\\ 0& =& \sum _{i=1}^n{x}_i{y}_i-n\overline{x}{\hat{\alpha }}_{ML}-{\hat{\beta }}_{ML}\sum _{i=1}^n{x}_i^2\\ 0& =& n-\frac{1}{{\sigma }_{ML}^2}\sum _{i=1}^n{(y_i-{\hat{\alpha }}_{ML}-{\hat{\beta }}_{ML}{x}_i)}^2\end{array}\end{array}$$

${\hat{\alpha }}_{ML}$を求める

$$\begin{array}{rcl}0& =& \overline{y}-{\hat{\alpha }}_{ML}-{\hat{\beta }}_{ML}\overline{x}\\ {\hat{\alpha }}_{ML}& =& \overline{y}-{\hat{\beta }}_{ML}\overline{x}\end{array}$$

${\hat{\beta }}_{ML}$を求める

$$\begin{array}{rcl}0& =& \sum _{i=1}^n{x}_i{y}_i-n\overline{x}{\hat{\alpha }}_{ML}-{\hat{\beta }}_{ML}\sum _{i=1}^n{x}_i^2\\ & =& \sum _{i=1}^n{x}_i{y}_i-n\overline{x}(\overline{y}-{\hat{\beta }}_{ML}\overline{x})-{\hat{\beta }}_{ML}\sum _{i=1}^n{x}_i^2\cdots {\hat{\alpha }}_{ML}=\overline{y}-{\hat{\beta }}_{ML}\overline{x}\\ & =& \sum _{i=1}^n{x}_i{y}_i-n\overline{x}\overline{y}+n{\hat{\beta }}_{ML}{\overline{x}}^2-{\hat{\beta }}_{ML}\sum _{i=1}^n{x}_i^2\\ & =& \sum _{i=1}^n{x}_i{y}_i-n\overline{x}\overline{y}-{\hat{\beta }}_{ML}\{\left(\sum _{i=1}^n{x}_i^2\right)-n{\overline{x}}^2\}\\ & =& \sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})-{\hat{\beta }}_{ML}\sum _{i=1}^n{(x_i-\overline{x})}^2\\ & =& S_{xy}-{\hat{\beta }}_{ML}{S}_{xx}\cdots S_{xy}=\sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y}),S_{xx}=\sum _{i=1}^n(x_i-\overline{x}{)}^2\\ {\hat{\beta }}_{ML}& =& \frac{S_{xy}}{S_{xx}}\\ & =& \hat{\beta }\cdots \mathrm{正}\mathrm{規}\mathrm{方}\mathrm{程}\mathrm{式}\mathrm{の}\mathrm{解}\mathrm{と}\mathrm{同}\mathrm{じ}\\ {\hat{\alpha }}_{ML}& =& \overline{y}-{\hat{\beta }}_{ML}\overline{x}\\ & =& \overline{y}-\hat{\beta }\overline{x}\\ & =& \hat{\alpha }\cdots \mathrm{正}\mathrm{規}\mathrm{方}\mathrm{程}\mathrm{式}\mathrm{の}\mathrm{解}\mathrm{と}\mathrm{同}\mathrm{じ}\end{array}$$

${\hat{\sigma }}_{ML}^2$を求める

$$\begin{array}{rcl}0& =& n-\frac{1}{{\sigma }_{ML}^2}\sum _{i=1}^n{(y_i-{\hat{\alpha }}_{ML}-{\hat{\beta }}_{ML}{x}_i)}^2\\ -n& =& -\frac{1}{{\sigma }_{ML}^2}\sum _{i=1}^n{(y_i-{\hat{\alpha }}_{ML}-{\hat{\beta }}_{ML}{x}_i)}^2\\ -n{\sigma }_{ML}^2& =& -\sum _{i=1}^n{(y_i-{\hat{\alpha }}_{ML}-{\hat{\beta }}_{ML}{x}_i)}^2\\ {\hat{\sigma }}_{ML}^2& =& \frac{1}{n}\sum _{i=1}^n{(y_i-{\hat{\alpha }}_{ML}-{\hat{\beta }}_{ML}{x}_i)}^2\\ & =& \frac{1}{n}\sum _{i=1}^n{(y_i-\hat{\alpha }-\hat{\beta }{x}_i)}^2\\ & =& \frac{1}{n}\sum _{i=1}^n{(y_i-{\hat{y}}_i)}^2\cdots {\hat{y}}_i=\hat{\alpha }+\hat{\beta }{x}_i\\ & =& \frac{1}{n}\sum _{i=1}^n{e}_i^2\\ & =& \frac{1}{n}(n-2)s^2\cdots \sum _{i=1}^n{e}_i^2=(n-2)s^2,s^2=\frac{1}{n-2}\sum _{i=1}^n{e}_i^2\\ & =& \frac{n-2}{n}{s}^2\end{array}$$

${\hat{\alpha }}_{ML},{\hat{\beta }}_{ML},{\hat{\sigma }}_{ML}^2$

以上より最尤推定量${\hat{\alpha }}_{ML},{\hat{\beta }}_{ML},{\hat{\sigma }}_{ML}^2$は以下のようになる． $$\begin{array}{rcl}{\hat{\beta }}_{ML}& =& \frac{S_{xy}}{S_{xx}}=\hat{\beta }\\ {\hat{\alpha }}_{ML}& =& \overline{y}-{\hat{\beta }}_{ML}\overline{x}=\overline{y}-\hat{\beta }\overline{x}=\hat{\alpha }\\ {\hat{\sigma }}_{ML}^2& =& \frac{n-2}{n}{s}^2\end{array}$$

確率変数の変数変換 Z=X/Y

確率変数の変数変換 Z=X/Y

Z=X/Y

$$\begin{array}{rcl}{p}_{X/Y}(z)& =& \int \int \delta (z-\frac{x}{y})f(x)g(y)\mathrm{d}x\mathrm{d}y\\ & & \cdots {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)f(x)\mathrm{d}x=f(0)\\ & & \cdots X=x,Y=y\mathrm{の}\mathrm{同}\mathrm{時}\mathrm{確}\mathrm{率}f(x)g(x)\mathrm{の}\mathrm{う}\mathrm{ち}z-\left(\frac{x}{y}\right)=0\mathrm{を}\mathrm{満}\mathrm{た}\mathrm{す}\mathrm{も}\mathrm{の}\mathrm{だ}\mathrm{け}\mathrm{を}\mathrm{足}\mathrm{し}\mathrm{合}\mathrm{わ}\mathrm{せ}\mathrm{る}.\\ & =& \int \int \left|y\right|\delta (x-yz)f\left(x\right)g(y)\mathrm{d}x\mathrm{d}y\\ & & \cdots \delta (u(x))=\sum _{\alpha \in u^{-1}(0)}\frac{1}{|u^{\prime }(\alpha )|}\delta (x-\alpha )\\ & & \cdots u(x)=z-\frac{x}{y}\\ & & \cdots u(x=\alpha )=0,\alpha =yz\\ & & \cdots u^{\prime }=\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{\mathrm{d}}{\mathrm{d}x}(z-\frac{x}{y})=-\frac{1}{y}\\ & & \cdots \delta (z-\frac{x}{y})=\frac{1}{|u^{\prime }(\alpha )|}\delta (x-\alpha )=\frac{1}{|-\frac{1}{y}|}\delta (x-yz)=\frac{1}{\left|\frac{1}{y}\right|}\delta (x-yz)=\left|y\right|\delta (x-yz)\\ & =& \int |y|\delta (yz-yz)f\left(yz\right)g(y)\mathrm{d}y\cdots z=\frac{x}{y},x=yz\\ & =& \int |y|\delta \left(0\right)f\left(yz\right)g(y)\mathrm{d}y\\ & =& \int |y|f(yz)g(y)\mathrm{d}y\cdots \delta \mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{引}\mathrm{数}\mathrm{が}\mathrm{全}\mathrm{域}\mathrm{で}0\mathrm{な}\mathrm{の}\mathrm{で}，\mathrm{全}\mathrm{域}\mathrm{で}|y|f(yz)g(y)\mathrm{を}\mathrm{足}\mathrm{し}\mathrm{合}\mathrm{わ}\mathrm{せ}\mathrm{る}\mathrm{積}\mathrm{分}\mathrm{に}\mathrm{な}\mathrm{る}.\end{array}$$

標準正規分布同士の例

$$\begin{array}{rcl}{f}_X(x)& =& \frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}\cdots N(0,1)(\mathrm{標}\mathrm{準}\mathrm{正}\mathrm{規}\mathrm{分}\mathrm{布})\\ g_Y(y)& =& \frac{1}{\sqrt{2\pi }}{e}^{-\frac{y^2}{2}}\cdots N(0,1)(\mathrm{標}\mathrm{準}\mathrm{正}\mathrm{規}\mathrm{分}\mathrm{布})\\ p_{X/Y}(z)& =& \int \int \delta (z-\frac{x}{y})f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y\\ & =& \int \int \delta (z-\frac{x}{y})\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{y^2}{2}}\mathrm{d}x\mathrm{d}y\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left|y\right|\frac{1}{\sqrt{2\pi }}{e}^{-\frac{{\left(yz\right)}^2}{2}}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{y^2}{2}}\mathrm{d}y\cdots z=\frac{x}{y},x=yz\\ & =& \frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left|y\right|e^{-\frac{{\left(yz\right)}^2}{2}}{e}^{-\frac{y^2}{2}}\mathrm{d}y\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left|y\right|e^{-\frac{{\left(yz\right)}^2}{2}-\frac{y^2}{2}}\mathrm{d}y\cdots A^B{A}^C=A^{B+C}\\ & =& \frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left|y\right|e^{-\frac{y^2(z^2+1)}{2}}\mathrm{d}y\\ & =& \frac{1}{2\pi }\{{\int }_{-\mathrm{\infty }}^0\left|y\right|e^{-\frac{y^2(z^2+1)}{2}}\mathrm{d}y+{\int }_0^{\mathrm{\infty }}\left|y\right|e^{-\frac{y^2(z^2+1)}{2}}\mathrm{d}y\}\\ & & \cdots {\int }_A^{B}f(x)\mathrm{d}x={\int }_A^{C}f(x)\mathrm{d}x+{\int }_C^{B}f(x)\mathrm{d}x(\mathrm{た}\mathrm{だ}\mathrm{し}A\le C\le B)\\ & =& \frac{1}{2\pi }\{{\int }_{-\mathrm{\infty }}^0-y{e}^{-\frac{y^2(z^2+1)}{2}}\mathrm{d}y+{\int }_0^{\mathrm{\infty }}y{e}^{-\frac{y^2(z^2+1)}{2}}\mathrm{d}y\}\cdots \left|A\right|=\{\begin{array}{}A& (A\ge 0)\\ -A& (A<0)\end{array}\\ & =& \frac{1}{2\pi }2{\int }_0^{\mathrm{\infty }}y{e}^{-\frac{y^2(z^2+1)}{2}}\mathrm{d}y\\ & & \cdots y=x\mathrm{が}\mathrm{奇}\mathrm{凾}\mathrm{数}\mathrm{で}y=e^{-\frac{x^2(z^2+1)}{2}}\mathrm{が}\mathrm{偶}\mathrm{凾}\mathrm{数}\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{そ}\mathrm{の}\mathrm{積}\mathrm{は}\mathrm{奇}\mathrm{凾}\mathrm{数}.\\ & & \cdots \mathrm{負}\mathrm{の}\mathrm{区}\mathrm{間}(-\mathrm{\infty },0)\mathrm{と}\mathrm{正}\mathrm{の}\mathrm{区}\mathrm{間}(0,\mathrm{\infty })\mathrm{で}\mathrm{符}\mathrm{号}\mathrm{が}\mathrm{逆}\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{全}\mathrm{体}\mathrm{と}\mathrm{し}\mathrm{て}\mathrm{は}\mathrm{偶}\mathrm{凾}\mathrm{数}\mathrm{と}\mathrm{同}\mathrm{じ}\mathrm{に}\mathrm{扱}\mathrm{え}\mathrm{る}.\\ & & \cdots -f_{\mathrm{奇}\mathrm{凾}\mathrm{数}(-\mathrm{\infty },0)}(x)+f_{\mathrm{奇}\mathrm{凾}\mathrm{数}(0,\mathrm{\infty })}(x)=f_{\mathrm{偶}\mathrm{凾}\mathrm{数}}(x)\mathrm{d}x\\ & & \cdots \mathrm{偶}\mathrm{凾}\mathrm{数}\mathrm{の}(-a,a)\mathrm{で}\mathrm{の}\mathrm{積}\mathrm{分}\mathrm{は}\mathrm{正}\mathrm{の}\mathrm{区}\mathrm{間}(0,a)\mathrm{の}2\mathrm{倍}\mathrm{と}\mathrm{し}\mathrm{て}\mathrm{計}\mathrm{算}\mathrm{で}\mathrm{き}\mathrm{る}.\\ & & \cdots {\int }_{-a}^a{f}_{\mathrm{奇}\mathrm{凾}\mathrm{数}}(x)\mathrm{d}x=2{\int }_0^a{f}_{\mathrm{偶}\mathrm{凾}\mathrm{数}}(x)\mathrm{d}x\\ & =& \frac{1}{\pi }{\int }_0^{\mathrm{\infty }}y{e}^{-\frac{y^2(z^2+1)}{2}}\mathrm{d}y\\ & =& \frac{1}{\pi }{\int }_0^{\mathrm{\infty }}y{e}^{-t}\cdot \frac{1}{(z^2+1)y}\mathrm{d}t\\ & & \cdots t=\frac{y^2(z^2+1)}{2},\frac{\mathrm{d}t}{\mathrm{d}y}=\frac{(z^2+1)}{2}2y=(z^2+1)y,\mathrm{d}y=\frac{1}{(z^2+1)y}\mathrm{d}t\\ & =& \frac{1}{\pi }\frac{1}{(z^2+1)}{\int }_0^{\mathrm{\infty }}{e}^{-t}\mathrm{d}t\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{\pi }\frac{1}{(z^2+1)}\cdot 1\cdots {\int }_0^{\mathrm{\infty }}{e}^{-t}\mathrm{d}t={[-e^{-t}]}_0^{\mathrm{\infty }}=[(-e^{-\mathrm{\infty }})-(-e^0)]=[0+1]=1\\ & =& \frac{1}{\pi }\frac{1}{(z^2+1)}\cdots \mathrm{標}\mathrm{準}\mathrm{コ}\mathrm{ー}\mathrm{シ}\mathrm{ー}\mathrm{分}\mathrm{布}\mathrm{に}\mathrm{等}\mathrm{し}\mathrm{い}.\\ f(x;x_0,\gamma )& =& \frac{1}{\pi }\frac{\gamma }{(x-x_0{)}^2+{\gamma }^2}\cdots \mathrm{コ}\mathrm{ー}\mathrm{シ}\mathrm{ー}\mathrm{分}\mathrm{布}(Cauchydistribution)\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{密}\mathrm{度}\mathrm{凾}\mathrm{数}\\ f(x;0,1)& =& \frac{1}{\pi }\frac{1}{(x-0{)}^2+1^2}\cdots \mathrm{標}\mathrm{準}\mathrm{コ}\mathrm{ー}\mathrm{シ}\mathrm{ー}\mathrm{分}\mathrm{布}\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{密}\mathrm{度}\mathrm{凾}\mathrm{数}\\ & =& \frac{1}{\pi }\frac{1}{x^2+1}\end{array}$$

指数分布同士の例

$$\begin{array}{rcl}{f}_X(x)& =& \lambda e^{-\lambda x}(x\ge 0,\lambda >0)\\ g_Y(y)& =& \lambda e^{-\lambda y}(x\ge 0,\lambda >0)\\ p_{X/Y}(z)& =& \int \int \delta (x-\frac{z}{y})f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y\\ & =& \int \int \delta (x-\frac{z}{y})\lambda e^{-\lambda x}\lambda e^{-\lambda y}\mathrm{d}x\mathrm{d}y\\ & =& {\int }_0^{\mathrm{\infty }}\left|y\right|\lambda e^{-\lambda yz}\lambda e^{-\lambda y}\mathrm{d}y\cdots z=\frac{x}{y},x=yz\\ & =& {\lambda }^2{\int }_0^{\mathrm{\infty }}\left|y\right|e^{-\lambda yz}{e}^{-\lambda y}\mathrm{d}y\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& {\lambda }^2{\int }_0^{\mathrm{\infty }}\left|y\right|e^{-\lambda yz-\lambda y}\mathrm{d}y\cdots A^B{A}^C=A^{B+C}\\ & =& {\lambda }^2{\int }_0^{\mathrm{\infty }}\left|y\right|e^{-\lambda y(z+1)}\mathrm{d}y\\ & =& {\lambda }^2{\int }_0^{\mathrm{\infty }}y{e}^{-\lambda y(z+1)}\mathrm{d}y\cdots y>0(\mathrm{積}\mathrm{分}\mathrm{範}\mathrm{囲}\mathrm{よ}\mathrm{り})\\ & =& {\lambda }^2{\int }_0^{\mathrm{\infty }}\frac{t}{\lambda (z+1)}{e}^{-t}\frac{1}{\lambda (z+1)}\mathrm{d}t\\ & & \cdots t=\lambda (z+1)y,y=\frac{t}{\lambda (z+1)}\\ & & \cdots \frac{\mathrm{d}t}{\mathrm{d}y}=\lambda (z+1),\mathrm{d}y=\frac{1}{\lambda (z+1)}\mathrm{d}t\\ & =& {\lambda }^2{\left\{\frac{1}{\lambda (z+1)}\right\}}^2{\int }_0^{\mathrm{\infty }}t{e}^{-t}\mathrm{d}t\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{{(z+1)}^2}{\int }_0^{\mathrm{\infty }}t{e}^{-t}\mathrm{d}t\\ & =& \frac{1}{{(z+1)}^2}{\int }_0^{\mathrm{\infty }}t{(-e^{-t})}^{\prime }\mathrm{d}t\cdots {(-e^{-t})}^{\prime }=e^{-t}\\ & =& \frac{1}{{(z+1)}^2}\{{[-t{e}^{-t}]}_0^{\mathrm{\infty }}-{\int }_0^{\mathrm{\infty }}-e^{-t}\mathrm{d}t\}\cdots {\int }_a^b{f}^{\prime }(x)g(x)\mathrm{d}x={[f(x)g(x)]}_a^b-{\int }_a^{b}f(x)g^{\prime }(x)\mathrm{d}x\\ & =& \frac{1}{{(z+1)}^2}\{[-(\mathrm{\infty })e^{-\mathrm{\infty }}-(-0e^{-0})]+{\int }_0^{\mathrm{\infty }}{e}^{-t}\mathrm{d}t\}\\ & =& \frac{1}{{(z+1)}^2}(0+1)\cdots {\int }_0^{\mathrm{\infty }}{e}^{-t}\mathrm{d}t={[-e^{-t}]}_0^{\mathrm{\infty }}=[(-e^{-\mathrm{\infty }})-(-e^0)]=[0+1]=1\\ & =& \frac{1}{{(z+1)}^2}\end{array}$$

確率変数の変数変換 Z=XY

確率変数の変数変換 Z=XY

Z=XY

$$\begin{array}{rcl}{p}_{XY}(z)& =& \int \int \delta (z-xy)f(x)g(y)\mathrm{d}x\mathrm{d}y\\ & & \cdots {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)f(x)\mathrm{d}x=f(0)\\ & & \cdots X=x,Y=y\mathrm{の}\mathrm{同}\mathrm{時}\mathrm{確}\mathrm{率}f(x)g(x)\mathrm{の}\mathrm{う}\mathrm{ち}z-(xy)=0\mathrm{を}\mathrm{満}\mathrm{た}\mathrm{す}\mathrm{も}\mathrm{の}\mathrm{だ}\mathrm{け}\mathrm{を}\mathrm{足}\mathrm{し}\mathrm{合}\mathrm{わ}\mathrm{せ}\mathrm{る}.\\ & =& \int \int \frac{1}{|y|}\delta (x-\frac{z}{y})f\left(x\right)g(y)\mathrm{d}x\mathrm{d}y\\ & & \cdots \delta (u(x))=\sum _{\alpha \in u^{-1}(0)}\frac{1}{|u^{\prime }(\alpha )|}\delta (x-\alpha )\\ & & \cdots u(x)=z-xy\\ & & \cdots u(x=\alpha )=0,\alpha =\frac{z}{y}\\ & & \cdots u^{\prime }=\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{\mathrm{d}}{\mathrm{d}x}(z-xy)=-y\\ & & \cdots \delta (z-xy)=\frac{1}{|u^{\prime }(\alpha )|}\delta (x-\alpha )=\frac{1}{|-y|}\delta (x-\frac{z}{y})=\frac{1}{|y|}\delta (x-\frac{z}{y})\\ & =& \int \frac{1}{|y|}\delta (\frac{z}{y}-\frac{z}{y})f\left(\frac{z}{y}\right)g(y)\mathrm{d}y\cdots z=xy,x=\frac{z}{y}\\ & =& \int \frac{1}{|y|}\delta \left(0\right)f\left(\frac{z}{y}\right)g(y)\mathrm{d}y\\ & =& \int \frac{1}{|y|}f\left(\frac{z}{y}\right)g(y)\mathrm{d}y\cdots \delta \mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{引}\mathrm{数}\mathrm{が}\mathrm{全}\mathrm{域}\mathrm{で}0\mathrm{な}\mathrm{の}\mathrm{で}，\mathrm{全}\mathrm{域}\mathrm{で}\frac{1}{|y|}f\left(\frac{z}{y}\right)g(y)\mathrm{を}\mathrm{足}\mathrm{し}\mathrm{合}\mathrm{わ}\mathrm{せ}\mathrm{る}\mathrm{積}\mathrm{分}\mathrm{に}\mathrm{な}\mathrm{る}.\end{array}$$

標準正規分布同士の例

$$\begin{array}{rcl}{f}_X(x)& =& \frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}\cdots N(0,1)(\mathrm{標}\mathrm{準}\mathrm{正}\mathrm{規}\mathrm{分}\mathrm{布})\\ g_Y(y)& =& \frac{1}{\sqrt{2\pi }}{e}^{-\frac{y^2}{2}}\cdots N(0,1)(\mathrm{標}\mathrm{準}\mathrm{正}\mathrm{規}\mathrm{分}\mathrm{布})\\ p_{XY}(z)& =& \int \int \delta (z-xy)f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y\\ & =& \int \int \delta (z-xy)\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{y^2}{2}}\mathrm{d}x\mathrm{d}y\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{|y|}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{{\left(\frac{z}{y}\right)}^2}{2}}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{y^2}{2}}\mathrm{d}y\cdots z=xy,x=\frac{z}{y}\\ & =& \frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{|y|}{e}^{-\frac{{\left(\frac{z}{y}\right)}^2}{2}}{e}^{-\frac{y^2}{2}}\mathrm{d}y\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{|y|}{e}^{-\frac{1}{2}(\frac{z^2}{y^2}+y^2)}\mathrm{d}y\\ & =& \frac{1}{2\pi }\{{\int }_{-\mathrm{\infty }}^0\frac{1}{|y|}{e}^{-\frac{1}{2}(\frac{z^2}{y^2}+y^2)}\mathrm{d}y+{\int }_0^{\mathrm{\infty }}\frac{1}{|y|}{e}^{-\frac{1}{2}(\frac{z^2}{y^2}+y^2)}\mathrm{d}y\}\\ & & \cdots {\int }_A^{B}f(x)\mathrm{d}x={\int }_A^{C}f(x)\mathrm{d}x+{\int }_C^{B}f(x)\mathrm{d}x(\mathrm{た}\mathrm{だ}\mathrm{し}A\le C\le B)\\ & =& \frac{1}{2\pi }\{{\int }_{-\mathrm{\infty }}^0-\frac{1}{y}{e}^{-\frac{1}{2}(\frac{z^2}{y^2}+y^2)}\mathrm{d}y+{\int }_0^{\mathrm{\infty }}\frac{1}{y}{e}^{-\frac{1}{2}(\frac{z^2}{y^2}+y^2)}\mathrm{d}y\}\\ & & \cdots \frac{1}{|y|}=-\frac{1}{y}(y<0),\frac{1}{|y|}=\frac{1}{y}(0<y)\\ & =& \frac{1}{2\pi }2{\int }_0^{\mathrm{\infty }}\frac{1}{y}{e}^{-\frac{1}{2}(\frac{z^2}{y^2}+y^2)}\mathrm{d}y\\ & & \cdots y=\frac{1}{x}\mathrm{が}\mathrm{奇}\mathrm{凾}\mathrm{数}\mathrm{で}y=e^{-\frac{1}{2}(\frac{z^2}{x^2}+x^2)}\mathrm{が}\mathrm{偶}\mathrm{凾}\mathrm{数}\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{そ}\mathrm{の}\mathrm{積}\mathrm{は}\mathrm{奇}\mathrm{凾}\mathrm{数}.\\ & & \cdots \mathrm{負}\mathrm{の}\mathrm{区}\mathrm{間}(-\mathrm{\infty },0)\mathrm{と}\mathrm{正}\mathrm{の}\mathrm{区}\mathrm{間}(0,\mathrm{\infty })\mathrm{で}\mathrm{符}\mathrm{号}\mathrm{が}\mathrm{逆}\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{全}\mathrm{体}\mathrm{と}\mathrm{し}\mathrm{て}\mathrm{は}\mathrm{偶}\mathrm{凾}\mathrm{数}\mathrm{と}\mathrm{同}\mathrm{じ}\mathrm{に}\mathrm{扱}\mathrm{え}\mathrm{る}.\\ & & \cdots -f_{\mathrm{奇}\mathrm{凾}\mathrm{数}(-\mathrm{\infty },0)}(x)+f_{\mathrm{奇}\mathrm{凾}\mathrm{数}(0,\mathrm{\infty })}(x)=f_{\mathrm{条}\mathrm{件}\mathrm{付}\mathrm{き}\mathrm{で}\mathrm{偶}\mathrm{凾}\mathrm{数}}(x)\mathrm{d}x\\ & & \cdots \mathrm{偶}\mathrm{凾}\mathrm{数}\mathrm{の}(-a,a)\mathrm{で}\mathrm{の}\mathrm{積}\mathrm{分}\mathrm{は}\mathrm{正}\mathrm{の}\mathrm{区}\mathrm{間}(0,a)\mathrm{の}2\mathrm{倍}\mathrm{と}\mathrm{し}\mathrm{て}\mathrm{計}\mathrm{算}\mathrm{で}\mathrm{き}\mathrm{る}.\\ & & \cdots {\int }_{-a}^a{f}_{\mathrm{奇}\mathrm{凾}\mathrm{数}}(x)\mathrm{d}x=2{\int }_0^a{f}_{\mathrm{偶}\mathrm{凾}\mathrm{数}}(x)\mathrm{d}x\\ & =& \frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\frac{1}{y}{e}^{-\frac{1}{2}(\frac{z^2}{y^2}+y^2)}\mathrm{d}y\\ & =& \frac{1}{\pi }{\int }_0^{\mathrm{\infty }}{e}^{-\frac{1}{2}\{\frac{z^2}{y^2}+y^2\}}\frac{\mathrm{d}t}{2t}\\ & & \cdots t=\frac{y^2}{2},y^2=2t,y:0\to \mathrm{\infty },t:0\to \mathrm{\infty }\\ & & \cdots \frac{\mathrm{d}t}{\mathrm{d}y}=y=\frac{y^2}{y}=\frac{2t}{y},\frac{\mathrm{d}y}{y}=\frac{\mathrm{d}t}{2t}\\ & =& \frac{1}{\pi }\frac{1}{2}{\int }_0^{\mathrm{\infty }}\frac{1}{t}{e}^{-\frac{1}{2}\{\frac{z^2}{2t}+2t\}}\mathrm{d}t\\ & =& \frac{1}{2\pi }{\int }_0^{\mathrm{\infty }}\frac{1}{t}{e}^{-t-\frac{z^2}{4t}}\mathrm{d}t\\ & & \cdots K_n(z)=\frac{1}{2}{\left(\frac{z}{2}\right)}^n{\int }_0^{\mathrm{\infty }}{t}^{-n-1}{e}^{-t-\frac{z^2}{4t}}\mathrm{d}t(\mathrm{第}\mathrm{二}\mathrm{種}\mathrm{変}\mathrm{形}\mathrm{ベ}\mathrm{ッ}\mathrm{セ}\mathrm{ル}\mathrm{凾}\mathrm{数})\\ & & \cdots K_0(z)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}{t}^{-1}{e}^{-t-\frac{z^2}{4t}}\mathrm{d}t\\ & =& \frac{1}{\pi }{K}_0(z)\end{array}$$

指数分布同士の例

$$\begin{array}{rcl}{f}_X(x)& =& \lambda e^{-\lambda x}(x\ge 0,\lambda >0)\\ g_Y(y)& =& \lambda e^{-\lambda y}(x\ge 0,\lambda >0)\\ p_{XY}(z)& =& \int \int \delta (x-xy)f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y\\ & =& \int \int \delta (x-xy)\lambda e^{-\lambda x}\lambda e^{-\lambda y}\mathrm{d}x\mathrm{d}y\\ & =& {\int }_0^{\mathrm{\infty }}\frac{1}{|y|}\lambda e^{-\lambda \left(\frac{z}{y}\right)}\lambda e^{-\lambda y}\mathrm{d}y\cdots z=xy,x=\frac{z}{y}\\ & =& {\lambda }^2{\int }_0^{\mathrm{\infty }}\frac{1}{|y|}{e}^{-\lambda \left(\frac{z}{y}\right)}{e}^{-\lambda y}\mathrm{d}y\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& {\lambda }^2{\int }_0^{\mathrm{\infty }}\frac{1}{|y|}{e}^{-\lambda \left(\frac{z}{y}\right)-\lambda y}\mathrm{d}y\cdots A^B{A}^C=A^{B+C}\\ & =& {\lambda }^2{\int }_0^{\mathrm{\infty }}\frac{1}{|y|}{e}^{-\lambda (\frac{z}{y}+y)}\mathrm{d}y\\ & =& {\lambda }^2{\int }_0^{\mathrm{\infty }}\frac{1}{|\frac{t}{\lambda }|}{e}^{-\lambda (\frac{z}{\frac{t}{\lambda }}+\frac{t}{\lambda })}\frac{1}{\lambda }\mathrm{d}t\\ & & \cdots t=\lambda y,y=\frac{t}{\lambda },\frac{\mathrm{d}y}{\mathrm{d}t}=\frac{1}{\lambda },\mathrm{d}y=\frac{1}{\lambda }\mathrm{d}t,y:0\to \mathrm{\infty },t:0\to \mathrm{\infty }\\ & =& {\lambda }^2{\int }_0^{\mathrm{\infty }}\frac{|\lambda |}{|t|}{e}^{-{\lambda }^2\frac{z}{t}-t}\frac{1}{\lambda }\mathrm{d}t\\ & =& {\lambda }^2{\int }_0^{\mathrm{\infty }}\frac{|\lambda |}{\lambda }\frac{1}{|t|}{e}^{-{\lambda }^2\frac{z}{t}-t}\mathrm{d}t\\ & =& {\lambda }^2{\int }_0^{\mathrm{\infty }}\frac{1}{|t|}{e}^{-{\lambda }^2\frac{z}{t}-t}\mathrm{d}t\cdots \lambda >0\\ & =& {\lambda }^2{\int }_0^{\mathrm{\infty }}\frac{1}{t}{e}^{-{\lambda }^2\frac{z}{t}-t}\mathrm{d}t\cdots t>0(\mathrm{積}\mathrm{分}\mathrm{範}\mathrm{囲}\mathrm{よ}\mathrm{り})\\ & =& 2{\lambda }^2{K}_0(2\lambda \sqrt{z})\\ & & \cdots K_n(z)=\frac{1}{2}{\left(\frac{z}{2}\right)}^n{\int }_0^{\mathrm{\infty }}{t}^{-n-1}{e}^{-t-\frac{z^2}{4t}}\mathrm{d}t(\mathrm{第}\mathrm{二}\mathrm{種}\mathrm{変}\mathrm{形}\mathrm{ベ}\mathrm{ッ}\mathrm{セ}\mathrm{ル}\mathrm{凾}\mathrm{数})\\ & & \cdots K_0(z)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}{t}^{-1}{e}^{-t-\frac{z^2}{4t}}\mathrm{d}t\end{array}$$

確率変数の変数変換 Z=X+Y

確率変数の変数変換 Z=X+Y

Z=X+Y

$$\begin{array}{rcl}{p}_{X+Y}(z)& =& \int \int \delta (z-(x+y))f(x)g(y)\mathrm{d}x\mathrm{d}y\\ & & \cdots {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)f(x)\mathrm{d}x=f(0)\\ & & \cdots X=x,Y=y\mathrm{の}\mathrm{同}\mathrm{時}\mathrm{確}\mathrm{率}f(x)g(x)\mathrm{の}\mathrm{う}\mathrm{ち}z-(x+y)=0\mathrm{を}\mathrm{満}\mathrm{た}\mathrm{す}\mathrm{も}\mathrm{の}\mathrm{だ}\mathrm{け}\mathrm{を}\mathrm{足}\mathrm{し}\mathrm{合}\mathrm{わ}\mathrm{せ}\mathrm{る}.\\ & =& \int \int \delta (z-((z-y)+y))f(z-y)g(y)\mathrm{d}y\cdots z=x+y,x=z-y\\ & =& \int \delta (0)f(z-y)g(y)\mathrm{d}y\\ & =& \int f(z-y)g(y)\mathrm{d}y\cdots \delta \mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{引}\mathrm{数}\mathrm{が}\mathrm{全}\mathrm{域}\mathrm{で}0\mathrm{な}\mathrm{の}\mathrm{で}，\mathrm{全}\mathrm{域}\mathrm{で}f(z-y)g(y)\mathrm{を}\mathrm{足}\mathrm{し}\mathrm{合}\mathrm{わ}\mathrm{せ}\mathrm{る}\mathrm{積}\mathrm{分}\mathrm{に}\mathrm{な}\mathrm{る}.\end{array}$$

標準正規分布同士の例

$$\begin{array}{rcl}{f}_X(x)& =& \frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}\cdots N(0,1)(\mathrm{標}\mathrm{準}\mathrm{正}\mathrm{規}\mathrm{分}\mathrm{布})\\ g_Y(y)& =& \frac{1}{\sqrt{2\pi }}{e}^{-\frac{y^2}{2}}\cdots N(0,1)(\mathrm{標}\mathrm{準}\mathrm{正}\mathrm{規}\mathrm{分}\mathrm{布})\\ p_{X+Y}(z)& =& \int \int \delta (z-(x+y))f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y\\ & =& \int \int \delta (z-(x+y))\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{y^2}{2}}\mathrm{d}x\mathrm{d}y\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{(z-y{)}^2}{2}}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{y^2}{2}}\mathrm{d}y\cdots z=x+y,x=z-y\\ & =& \frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-\frac{(z-y{)}^2}{2}}{e}^{-\frac{y^2}{2}}\mathrm{d}y\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{2\pi }\int e^{-\frac{(z-y{)}^2}{2}-\frac{y^2}{2}}\mathrm{d}y\cdots A^B{A}^C=A^{B+C}\\ & =& \frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-\frac{(z-y{)}^2+y^2}{2}}\mathrm{d}y\\ & =& \frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-\frac{z^2-2yz+y^2+y^2}{2}}\mathrm{d}y\\ & =& \frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-\frac{z^2-2yz+2y^2}{2}}\mathrm{d}y\\ & =& \frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-\frac{z^2}{2}+yz-y^2}\mathrm{d}y\\ & =& \frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-\frac{z^2}{2}+\frac{z^2}{4}-\frac{z^2}{4}+yz-y^2\mathrm{d}y}\\ & =& \frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-\frac{z^2}{4}-(\frac{z^2}{4}-yz+y^2)}\mathrm{d}y\\ & =& \frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-\frac{z^2}{4}}{e}^{-(\frac{z^2}{4}-yz+y^2)}\mathrm{d}y\\ & =& \frac{1}{2\pi }{e}^{-\frac{z^2}{4}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-(\frac{z}{2}-y{)}^2}\mathrm{d}y\\ & =& \frac{1}{2\pi }{e}^{-\frac{z^2}{4}}{\int }_{\mathrm{\infty }}^{-\mathrm{\infty }}{e}^{-t^2}(-1)\mathrm{d}t\\ & & \cdots t=\frac{z}{2}-y,\frac{\mathrm{d}t}{\mathrm{d}y}=-1,\mathrm{d}y=-\mathrm{d}t,y:-\mathrm{\infty }\to \mathrm{\infty },t:\mathrm{\infty }\to -\mathrm{\infty }\\ & =& \frac{1}{2\pi }{e}^{-\frac{z^2}{4}}(-1){\int }_{\mathrm{\infty }}^{-\mathrm{\infty }}{e}^{-t^2}\mathrm{d}t\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{2\pi }{e}^{-\frac{z^2}{4}}(-1)(-{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-t^2}\mathrm{d}t)\cdots {\int }_A^{B}f(x)\mathrm{d}x=-{\int }_B^{A}f(x)\mathrm{d}x\\ & =& \frac{1}{2\pi }{e}^{-\frac{z^2}{4}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-t^2}\mathrm{d}t\\ & =& \frac{1}{2\pi }{e}^{-\frac{z^2}{4}}\sqrt{\pi }\cdots {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-t^2}\mathrm{d}y=\sqrt{\pi }\\ & =& \frac{1}{2\sqrt{\pi }}{e}^{-\frac{z^2}{4}}\\ & =& \frac{1}{\sqrt{2\pi \cdot 2}}{e}^{-\frac{z^2}{2\cdot 2}}\cdots N(0,2)\end{array}$$

指数分布同士の例

$$\begin{array}{rcl}{f}_X(x)& =& \lambda e^{-\lambda x}(x\ge 0,\lambda >0)\cdots Exp(\lambda )(\mathrm{指}\mathrm{数}\mathrm{分}\mathrm{布})\\ g_Y(y)& =& \lambda e^{-\lambda y}(x\ge 0,\lambda >0)\cdots Exp(\lambda )(\mathrm{指}\mathrm{数}\mathrm{分}\mathrm{布})\\ p_{X+Y}(z)& =& \int \int \delta (z-(x+y))f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y\\ & =& \int \int \delta (z-(x+y))\lambda e^{-\lambda x}\lambda e^{-\lambda y}\mathrm{d}x\mathrm{d}y\\ & =& {\int }_0^z\lambda e^{-\lambda (z-y)}\lambda e^{-\lambda y}\mathrm{d}y\cdots z=x+y,x=z-y\\ & =& {\lambda }^2{\int }_0^z{e}^{-\lambda (z-y)}{e}^{-\lambda y}\mathrm{d}y\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& {\lambda }^2{\int }_0^z{e}^{-\lambda (z-y)-\lambda y}\mathrm{d}y\cdots A^B{A}^C=A^{B+C}\\ & =& {\lambda }^2{\int }_0^z{e}^{-\lambda (z-y+y)}\mathrm{d}y\\ & =& {\lambda }^2{\int }_0^z{e}^{-\lambda z}\mathrm{d}y\\ & =& {\lambda }^2{e}^{-\lambda z}{\int }_0^z\mathrm{d}y\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& {\lambda }^2{e}^{-\lambda z}{\left[y\right]}_0^z\\ & =& {\lambda }^2{e}^{-\lambda z}[z-0]\\ & =& {\lambda }^{2}z{e}^{-\lambda z}\\ & =& \mathrm{\Gamma }(2,\lambda )\end{array}$$ $$\begin{array}{rcl}\mathrm{\Gamma }(\alpha ,\lambda )& =& \frac{1}{\mathrm{\Gamma }(\alpha )}{\lambda }^{\alpha }{x}^{\alpha -1}{e}^{-\lambda x}(\mathrm{ガ}\mathrm{ン}\mathrm{マ}\mathrm{分}\mathrm{布};gammadisribution)\\ \mathrm{\Gamma }(1,\lambda )& =& \frac{1}{\mathrm{\Gamma }(1)}{\lambda }^1{x}^{1-1}{e}^{-\lambda x}=\frac{1}{1}\lambda e^{-\lambda x}=\lambda e^{-\lambda x}\\ & & \cdots \mathrm{\Gamma }(1)=(1-1)!=0!=1\\ & & \cdots \mathrm{指}\mathrm{数}\mathrm{分}\mathrm{布}\\ \mathrm{\Gamma }(2,\lambda )& =& \frac{1}{\mathrm{\Gamma }(2)}{\lambda }^2{x}^{2-1}{e}^{-\lambda x}=\frac{1}{1}{\lambda }^{2}x{e}^{-\lambda x}={\lambda }^{2}x{e}^{-\lambda x}\\ & & \cdots \mathrm{\Gamma }(2)=(2-1)!=1!=1\\ & & \cdots \mathrm{指}\mathrm{数}\mathrm{分}\mathrm{布}\mathrm{に}\mathrm{従}\mathrm{う}\mathrm{確}\mathrm{率}\mathrm{変}\mathrm{数}\mathrm{同}\mathrm{士}\mathrm{の}\mathrm{和}\mathrm{の}\mathrm{分}\mathrm{布},\alpha =2\mathrm{の}\mathrm{ガ}\mathrm{ン}\mathrm{マ}\mathrm{分}\mathrm{布}\end{array}$$

デルタ凾数(delta function)

デルタ凾数(delta function)

以下の式を満たす$\delta (x)$をデルタ凾数と呼ぶ． $$\begin{array}{rcl}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)f(x)\mathrm{d}x& =& f(0)\end{array}$$ これは$f(x)$の積分の中から$x=0(\delta (0))$となる時のf(x)のみを取り出すことになる.

上記式を満たすため，$\delta (x)$のみでの積分値は1でとされる． $$\begin{array}{rcl}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)\mathrm{d}x& =& 1\cdots \mathrm{前}\mathrm{述}\mathrm{の}\mathrm{式}\mathrm{に}\mathrm{お}\mathrm{い}\mathrm{て}f(x)=1\mathrm{の}\mathrm{場}\mathrm{合}\mathrm{と}\mathrm{同}\mathrm{じ}\end{array}$$

デルタ凾数の性質

$\delta (x-\alpha )$の場合

足し合わせたいxの値を動かす．これは$f(x)$の積分の中から$x=\alpha (\delta (\alpha -\alpha )=\delta (0))$となる時の$f(\alpha )$のみを取り出すことになる. $$\begin{array}{r}\\ {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x-\alpha )f(x)\mathrm{d}x& =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (t)f(t+\alpha )\mathrm{d}t\\ & & \cdots t=x-\alpha ,x=t+\alpha ,\frac{\mathrm{d}x}{\mathrm{d}t}=1,\mathrm{d}x=\mathrm{d}t\\ & & \cdots x:-\mathrm{\infty }\to \mathrm{\infty },t:-\mathrm{\infty }\to \mathrm{\infty }\\ & =& f(\alpha )\end{array}$$

$\delta (\alpha x)$の場合

$f(x)$と$\delta (x)$でスケールが$\alpha$倍異なる場合の性質を求める． $$\begin{array}{rcl}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (\alpha x)f(x)\mathrm{d}x& =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (t)f\left(\frac{t}{\alpha }\right)\frac{\mathrm{d}t}{\alpha }\\ & & \cdots t=\alpha x,x=\frac{t}{\alpha }\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{1}{\alpha },\mathrm{d}x=\frac{\mathrm{d}t}{\alpha }\\ & & \cdots x:-\mathrm{\infty }\to \mathrm{\infty },t:-\mathrm{\infty }\to \mathrm{\infty }\\ & =& \{\begin{array}{ll}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (t)f\left(\frac{t}{\alpha }\right)\frac{\mathrm{d}t}{|\alpha |}& (\alpha >0)\cdots a=|a|(a>0)\\ {\int }_{\mathrm{\infty }}^{-\mathrm{\infty }}\delta (t)f\left(\frac{t}{\alpha }\right)\frac{\mathrm{d}t}{|\alpha |}& (\alpha <0)\cdots a=-|a|(a<0)\end{array}\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (t)f\left(\frac{t}{\alpha }\right)\frac{\mathrm{d}t}{|\alpha |}\\ & & \cdots t=0\mathrm{以}\mathrm{外}\mathrm{は}\delta (t)=0\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{積}\mathrm{分}\mathrm{範}\mathrm{囲}\mathrm{が}\mathrm{逆}\mathrm{転}\mathrm{し}\mathrm{て}\mathrm{も}\delta (t)=0\mathrm{で}\mathrm{の}\\ & & \cdots f(t)\mathrm{の}\mathrm{値}\mathrm{以}\mathrm{外}\mathrm{関}\mathrm{係}\mathrm{な}\mathrm{く},\alpha \mathrm{の}\mathrm{絶}\mathrm{対}\mathrm{値}\mathrm{を}\mathrm{と}\mathrm{れ}\mathrm{ば}\alpha \mathrm{の}\mathrm{正}\mathrm{負}\mathrm{に}\mathrm{よ}\mathrm{ら}\mathrm{な}\mathrm{い}\mathrm{式}\mathrm{に}\mathrm{で}\mathrm{き}\mathrm{る}．\\ & =& \frac{1}{|\alpha |}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (t)f\left(\frac{t}{\alpha }\right)\mathrm{d}t\\ & =& \frac{1}{|\alpha |}f\left(0\right)\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{|\alpha |}\delta (t)f\left(x\right)\mathrm{d}t\cdots \mathrm{最}\mathrm{初}\mathrm{の}\mathrm{式}\mathrm{と}\mathrm{比}\mathrm{較}\mathrm{し}\mathrm{や}\mathrm{す}\mathrm{い}\mathrm{よ}\mathrm{う}\mathrm{に}\mathrm{並}\mathrm{べ}\mathrm{直}\mathrm{し}\mathrm{て}\mathrm{み}\mathrm{る}．\\ \delta (\alpha x)& =& \frac{1}{|\alpha |}\delta (t)\end{array}$$

$\delta (u(x))$の場合

足し合わせたいxの値を動かす．$u(x)=0$を満たす$x$を$x={\alpha }_1,\cdots ,{\alpha }_k,\cdots ,{\alpha }_n$とするとこれは$f(x)$の積分の中から$x={\alpha }_k(\delta ({\alpha }_k-{\alpha }_k)=\delta (0))$となる時の$f({\alpha }_k)$のみを取り出すことになり，積分はその足し合わせとなる.

$u(x)=0$の解$x={\alpha }_k$が区間にただ一つとなるように$ϵ$を十分小さくとるとする． $$\begin{array}{rcl}{\int }_{{\alpha }_k-ϵ}^{{\alpha }_k+ϵ}\delta (u(x))f(x)\mathrm{d}x& =& {\int }_{{\alpha }_k-ϵ}^{{\alpha }_k+ϵ}\delta (u({\alpha }_k)+u^{\prime }({\alpha }_k)(x-{\alpha }_k)+\frac{1}{2!}{u}^{\prime \prime }({\alpha }_k){(x-{\alpha }_k)}^2+\cdots )f(x)\mathrm{d}x\\ & & \cdots u(x)\mathrm{の}x=\alpha \mathrm{周}\mathrm{り}\mathrm{で}\mathrm{の}\mathrm{テ}\mathrm{イ}\mathrm{ラ}\mathrm{ー}\mathrm{展}\mathrm{開}\mathrm{を}\mathrm{行}\mathrm{っ}\mathrm{た}．\\ & & \cdots a\mathrm{周}\mathrm{り}\mathrm{で}\mathrm{の}f(x)\mathrm{の}\mathrm{テ}\mathrm{イ}\mathrm{ラ}\mathrm{ー}\mathrm{展}\mathrm{開}:f(x)=f^{}(a)+f^{\prime }(a)(x-a)+\frac{1}{2}{f}^{\prime \prime }(a){(x-a)}^2+\cdots \\ & =& {\int }_{{\alpha }_k-ϵ}^{{\alpha }_k+ϵ}\delta (0+u^{\prime }({\alpha }_k)(x-{\alpha }_k)+0)f(x)\mathrm{d}x\\ & & \cdots u({\alpha }_k)\mathrm{は}{\alpha }_k\mathrm{の}\mathrm{定}\mathrm{義}\mathrm{よ}\mathrm{り}0\mathrm{で}\mathrm{あ}\mathrm{る}．\\ & & \cdots (x-{\alpha }_k)=\pm ϵ\mathrm{で}\mathrm{あ}\mathrm{り}\mathrm{二}\mathrm{次}\mathrm{以}\mathrm{上}\mathrm{の}\mathrm{項}\mathrm{は}{ϵ}^2\mathrm{以}\mathrm{下}\mathrm{の}\mathrm{数}\mathrm{を}\mathrm{掛}\mathrm{け}\mathrm{る}\mathrm{こ}\mathrm{と}\mathrm{に}\mathrm{な}\mathrm{る}．\\ & & \cdots \mathrm{こ}\mathrm{れ}\mathrm{は}\mathrm{無}\mathrm{視}\mathrm{で}\mathrm{き}\mathrm{る}\mathrm{ほ}\mathrm{ど}\mathrm{小}\mathrm{さ}\mathrm{く},0\mathrm{と}\mathrm{し}\mathrm{て}\mathrm{扱}\mathrm{え}\mathrm{る}.\\ & =& {\int }_{{\alpha }_k-ϵ}^{{\alpha }_k+ϵ}\delta (u^{\prime }({\alpha }_k)(x-{\alpha }_k))f(x)\mathrm{d}x\\ & =& {\int }_{{\alpha }_k-ϵ}^{{\alpha }_k+ϵ}\frac{1}{|u^{\prime }({\alpha }_k)|}\delta (x-{\alpha }_k)f(x)\mathrm{d}x\cdots \delta (\alpha x)=\frac{1}{|\alpha |}\delta (t)(\mathrm{前}\mathrm{述})\end{array}$$ よって全区間では以下のようになる． $$\begin{array}{rcl}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (u(x))f(x)\mathrm{d}x& =& \sum _{k=1}^n{\int }_{{\alpha }_k-ϵ}^{{\alpha }_k+ϵ}\delta (u(x))f(x)\mathrm{d}x\\ & =& \sum _{k=1}^n{\int }_{{\alpha }_k-ϵ}^{{\alpha }_k+ϵ}\frac{1}{|u^{\prime }({\alpha }_k)|}\delta (x-{\alpha }_k)f(x)\mathrm{d}x\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left\{\sum _{k=1}^n\frac{1}{|u^{\prime }({\alpha }_k)|}\delta (x-{\alpha }_k)\right\}f(x)\mathrm{d}x\end{array}$$ $$\begin{array}{r}\\ \delta (u(x))& =& \sum _{k=1}^n\frac{1}{|u^{\prime }({\alpha }_k)|}\delta (x-{\alpha }_k)(\mathrm{た}\mathrm{だ}\mathrm{し}{\alpha }_k\mathrm{は}u(x)=0\mathrm{の}\mathrm{解})\\ & =& \sum _{\alpha \in u^{-1}(0)}\frac{1}{|u^{\prime }(\alpha )|}\delta (x-\alpha )\end{array}$$

確率母凾数 ( probability generating function )

確率母凾数

非負の整数値をとる離散型確率変数$X$に対して以下のように確率母凾数(probability generating function;積率母凾数ではない)が定義される． $$\begin{array}{rcl}{G}_X(t)& =& \mathrm{E}\left[t^X\right]\cdots \mathrm{積}\mathrm{率}\mathrm{母}\mathrm{凾}\mathrm{数}\mathrm{は}{M}_X(t)=\mathrm{E}\left[e^{tX}\right]\\ & =& \sum _k{t}^{k}P(X=k)\\ & & \cdots P(X):\mathrm{確}\mathrm{率}\mathrm{質}\mathrm{量}\mathrm{凾}\mathrm{数},\sum _k:X\mathrm{の}\mathrm{定}\mathrm{義}\mathrm{範}\mathrm{囲}\mathrm{す}\mathrm{べ}\mathrm{て}\mathrm{の}k\mathrm{で}\mathrm{の}\mathrm{和}\end{array}$$

一階微分((原点周りの)一次モーメント) = 期待値

$$\begin{array}{rcl}{G_X^{(1)}(t)|}_{t=1}& =& {\frac{\mathrm{d}}{\mathrm{d}t}{G}_X(t)|}_{t=1}\\ & =& {\sum _{k\ge 1}k{t}^{k-1}P(X=k)|}_{t=1}\\ & =& \sum _{k\ge 1}k{1}^{k-1}P(X=k)\\ & =& \sum _{k\ge 1}kP(X=k)\\ & =& \mathrm{E}\left[X\right]\end{array}$$

二階微分((原点周りの)二次モーメント)

$$\begin{array}{rcl}{G_X^{(2)}(t)|}_{t=1}& =& {\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}{G}_X(t)|}_{t=1}\\ & =& {\frac{\mathrm{d}}{\mathrm{d}t}\sum _{k\ge 1}k{t}^{k-1}P(X=k)|}_{t=1}\\ & =& {\sum _{k\ge 2}k(k-1)t^{k-2}P(X=k)|}_{t=1}\\ & =& \sum _{k\ge 2}k(k-1)1^{k-2}P(X=k)\\ & =& \sum _{k\ge 2}k(k-1)P(X=k)\\ & =& \mathrm{E}[X(X-1)]\end{array}$$

分散((母平均周りの)二次モーメント)

$$\begin{array}{rcl}\mathrm{V}\left[X\right]& =& \mathrm{E}\left[{(X-\mathrm{E}\left[X\right])}^2\right]\\ & =& \mathrm{E}\left[X^2\right]-\mathrm{E}{\left[X\right]}^2\cdots \mathrm{E}\left[{(X-\mathrm{E}\left[X\right])}^2\right]=\mathrm{E}\left[X^2\right]-{\left[X\right]}^2\\ & =& \mathrm{E}\left[X^2\right]-\mathrm{E}\left[X\right]+\mathrm{E}\left[X\right]-\mathrm{E}{\left[X\right]}^2\\ & =& \mathrm{E}[X^2-X]+\mathrm{E}\left[X\right]-\mathrm{E}{\left[X\right]}^2\cdots \mathrm{E}[X\pm Y]=\mathrm{E}\left[X\right]\pm \mathrm{E}\left[Y\right]\\ & =& \mathrm{E}[X(X-1)]+\mathrm{E}\left[X\right]-\mathrm{E}{\left[X\right]}^2\\ & =& \left({G_X^{(2)}(t)|}_{t=1}\right)+\left({G_X^{(1)}(t)|}_{t=1}\right)-{\left({G_X^{(1)}(t)|}_{t=1}\right)}^2\end{array}$$

Z=X+Y

$$\begin{array}{rcl}{G}_Z(t)& =& \mathrm{E}\left[t^Z\right]\\ & =& \mathrm{E}\left[t^{X+Y}\right]\\ & =& \mathrm{E}\left[t^X{t}^Y\right]\cdots A^{B+C}=A^B{A}^C\\ & =& \mathrm{E}\left[t^X\right]\mathrm{E}\left[t^Y\right]\cdots \mathrm{E}\left[AB\right]=\mathrm{E}\left[A\right]\mathrm{E}\left[B\right]A\mathrm{と}B\mathrm{が}\mathrm{独}\mathrm{立}\mathrm{の}\mathrm{場}\mathrm{合}\\ & =& G_X(t)G_Y(t)\end{array}$$

二項分布(binomial distribution)の積率母凾数(moment-generating function)と期待値(expected value)・分散(variance)

二項分布(binomial distribution)の積率母凾数(moment-generating function)と期待値(expected value)・分散(variance)

二項分布

$$\begin{array}{rcl}X& \sim & B(n,p)\\ f(X=x)& =& \{\begin{array}{ll}{}_n{\mathrm{C}}_x{p}^x(1-p{)}^{n-x}& x\in \{0,1,2,\dots ,n\}\\ 0& x\notin \{0,1,2,\dots ,n\}\end{array}\cdots \mathrm{確}\mathrm{率}\mathrm{密}\mathrm{度}\mathrm{凾}\mathrm{数}\end{array}$$

積率母凾数

$$\begin{array}{rcl}{M}_X(t)& =& \mathrm{E}\left[e^{tx}\right]\\ & =& \sum _{k=0}^n{e}^{tx}{}_n{\mathrm{C}}_x{p}^x{(1-p)}^{n-x}\\ & =& \sum _{k=0}^n{}_n{\mathrm{C}}_x{\left(e^{t}p\right)}^x{(1-p)}^{n-x}\\ & =& {}_n{\mathrm{C}}_0{\left(e^{t}p\right)}^0{(1-p)}^{n-0}+{}_n{\mathrm{C}}_1{\left(e^{t}p\right)}^1{(1-p)}^{n-1}+{}_n{\mathrm{C}}_2{\left(e^{t}p\right)}^2{(1-p)}^{n-2}+\cdots +{}_n{\mathrm{C}}_n{\left(e^{t}p\right)}^n{(1-p)}^{n-n}\\ & =& {\{e^{t}p+(1-p)\}}^n\\ & & \cdots (A+B{)}^D={}_D{\mathrm{C}}_0{A}^0{B}^{D-0}+{}_D{\mathrm{C}}_1{A}^1{B}^{D-1}+\cdots +{}_D{\mathrm{C}}_D{A}^D{B}^{D-D}=\sum _{k=0}^D{}_D{\mathrm{C}}_k{A}^k{B}^{D-k}(\mathrm{二}\mathrm{項}\mathrm{関}\mathrm{係})\\ & =& {(e^{t}p-p+1)}^n\end{array}$$

(原点周りの)一次モーメント = 期待値

$$\begin{array}{rcl}\mathrm{E}\left[X\right]& =& M_X^{(1)}(t)\\ & =& {\frac{\mathrm{d}{M}_X(t)}{\mathrm{d}t}|}_{t=0}\\ & =& {\frac{\mathrm{d}}{\mathrm{d}t}{(e^{t}p-p+1)}^n|}_{t=0}\\ & =& {\frac{\mathrm{d}}{\mathrm{d}u}{u}^n\frac{\mathrm{d}u}{\mathrm{d}t}|}_{t=0}\cdots u=e^{t}p-p+1,\frac{\mathrm{d}u}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}t}(e^{t}p-p+1)=e^{t}p\\ & =& {n{u}^{n-1}{e}^{t}p|}_{t=0}\\ & =& {n{(e^{t}p-p+1)}^{n-1}{e}^{t}p|}_{t=0}\\ & =& {np{e}^t{(e^{t}p-p+1)}^{n-1}|}_{t=0}\\ & =& np{e}^0{(e^{0}p-p+1)}^{n-1}\\ & =& np\cdot 1{(1\cdot p-p+1)}^{n-1}\\ & =& np{\left(1\right)}^{n-1}\\ & =& np\cdots X\mathrm{の}\mathrm{期}\mathrm{待}\mathrm{値}\mathrm{の}\mathrm{定}\mathrm{義}\mathrm{か}\mathrm{ら}\mathrm{求}\mathrm{め}\mathrm{た}\mathrm{二}\mathrm{項}\mathrm{分}\mathrm{布}\mathrm{の}\mathrm{期}\mathrm{待}\mathrm{値}\mathrm{と}\mathrm{同}\mathrm{じ}\end{array}$$

(原点周りの)二次モーメント

$$\begin{array}{rcl}\mathrm{E}\left[X^2\right]& =& M_X^{(2)}(t)\\ & =& {\frac{{\mathrm{d}}^2{M}_X(t)}{\mathrm{d}{t}^2}|}_{t=0}\\ & =& {\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}{(e^{t}p-p+1)}^n|}_{t=0}\\ & =& {\frac{\mathrm{d}}{\mathrm{d}t}np{e}^t{(e^{t}p-p+1)}^{n-1}|}_{t=0}\\ & =& {np\frac{\mathrm{d}}{\mathrm{d}t}{e}^t{(e^{t}p-p+1)}^{n-1}|}_{t=0}\cdots \frac{\mathrm{d}}{\mathrm{d}x}cf(x)=c\frac{\mathrm{d}}{\mathrm{d}x}f(x)(c:\mathrm{定}\mathrm{数})\\ & =& {np\frac{\mathrm{d}}{\mathrm{d}t}uv|}_{t=0}\cdots u=e^t,v={(e^{t}p-p+1)}^{n-1}\\ & =& {np\{\left(\frac{\mathrm{d}}{\mathrm{d}t}u\right)v+u\left(\frac{\mathrm{d}}{\mathrm{d}t}v\right)\}|}_{t=0}\\ & =& {np[\left(e^t\right)v+u\left\{(n-1){(e^{t}p-p+1)}^{n-2}p{e}^t\right\}]|}_{t=0}\cdots \frac{\mathrm{d}u}{\mathrm{d}t}=e^t,\frac{\mathrm{d}v}{\mathrm{d}t}=(n-1){(e^{t}p-p+1)}^{n-2}p{e}^t\\ & =& {np[\left(e^t\right){(e^{t}p-p+1)}^{n-1}+e^t\left\{(n-1){(e^{t}p-p+1)}^{n-2}p{e}^t\right\}]|}_{t=0}\cdots u=e^t,v={(e^{t}p-p+1)}^{n-1}\\ & =& {np{e}^t{(e^{t}p-p+1)}^{n-1}+n(n-1)p^2{e}^{2t}{(e^{t}p-p+1)}^{n-2}|}_{t=0}\\ & =& np{e}^0{(e^{0}p-p+1)}^{n-1}+n(n-1)p^2{e}^{2\cdot 0}{(e^{0}p-p+1)}^{n-2}\\ & =& np\cdot 1\cdot {(1\cdot p-p+1)}^{n-1}+n(n-1)p^2\cdot 1\cdot {(1\cdot p-p+1)}^{n-2}\\ & =& np{\left(1\right)}^{n-1}+n(n-1)p^2{\left(1\right)}^{n-2}\\ & =& np+n(n-1)p^2\end{array}$$

二次の中心(化)モーメント / 母平均周りの二次モーメント = 分散

$$\begin{array}{rcl}\mathrm{V}\left[X\right]& =& \mathrm{E}\left[{(X-\mathrm{E}\left[X\right])}^2\right]\\ & =& \mathrm{E}\left[X^2\right]-\mathrm{E}{\left[X\right]}^2\cdots \mathrm{E}\left[{(X-\mathrm{E}\left[X\right])}^2\right]=\mathrm{E}\left[X^2\right]-{\left[X\right]}^2\\ & =& M_X^{(2)}(t)-{(M_X^{(1)}(t))}^2\\ & =& np+n(n-1)p^2-(np{)}^2\\ & =& np+n^2{p}^2-n{p}^2-n^2{p}^2\\ & =& np-n{p}^2\\ & =& np(1-p)\cdots X(X-1)\mathrm{の}\mathrm{期}\mathrm{待}\mathrm{値}\mathrm{を}\mathrm{利}\mathrm{用}\mathrm{し}\mathrm{た}\mathrm{二}\mathrm{項}\mathrm{分}\mathrm{布}\mathrm{の}\mathrm{分}\mathrm{散}\mathrm{と}\mathrm{同}\mathrm{じ}\end{array}$$

正規分布に従う互いに独立な標本における分散の最尤推定量を求める

正規分布に従う互いに独立な標本における分散の最尤推定量を求める

$x_1,\cdots ,x_n$を$N(\mu ,{\sigma }^2)$からの独立な観測値とする時の${\sigma }^2$の最尤推定を考える．

確率密度凾数

$$\begin{array}{rcl}{X}_k& \sim & N(\mu ,{\sigma }^2)\\ f(x;\mu ,{\sigma }^2)& =& \frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}\cdots \mathrm{正}\mathrm{規}\mathrm{分}\mathrm{布}\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{密}\mathrm{度}\mathrm{凾}\mathrm{数}\\ f(x_1,\cdots ,x_n;\mu ,{\sigma }^2)& =& \left\{\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x_1-\mu {)}^2}{2{\sigma }^2}}\right\}\left\{\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x_2-\mu {)}^2}{2{\sigma }^2}}\right\}\cdots \left\{\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x_n-\mu {)}^2}{2{\sigma }^2}}\right\}\\ & =& {\left(\frac{1}{\sqrt{2\pi {\sigma }^2}}\right)}^n{e}^{-\frac{1}{2{\sigma }^2}\sum _{k=1}^n(x_k-\mu {)}^2}\end{array}$$

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$\mu$が既知の場合の${\sigma }^2$の最尤推定

確率密度凾数から尤度凾数(確率密度凾数に対して，確率変数や既知のパラメータを定数として，未知のパラメータを変数とみなす)の対数をとり，対数尤度凾数を用意する． $$\begin{array}{r}\\ l({\sigma }^2;x_1,\cdots ,x_n,\mu )& =& \log \left\{{\left(\frac{1}{\sqrt{2\pi {\sigma }^2}}\right)}^n{e}^{-\frac{1}{2{\sigma }^2}\sum _{k=1}^n(x_k-\mu {)}^2}\right\}\\ & =& \log \left\{{\left(\frac{1}{\sqrt{2\pi {\sigma }^2}}\right)}^n\right\}+\log \left\{e^{-\frac{1}{2{\sigma }^2}\sum _{k=1}^n(x_k-\mu {)}^2}\right\}\cdots \log \left(AB\right)=\log \left(A\right)+\log \left(B\right)\\ & =& \log \left\{{\left(2\pi {\sigma }^2\right)}^{-\frac{n}{2}}\right\}+\log \left\{e^{-\frac{1}{2{\sigma }^2}\sum _{k=1}^n(x_k-\mu {)}^2}\right\}\cdots \frac{1}{A}=A^{-1}\\ & =& -\frac{n}{2}\log \left(2\pi {\sigma }^2\right)-\frac{1}{2{\sigma }^2}\sum _{k=1}^n(x_k-\mu {)}^2\log \left(e\right)\cdots \log \left(A^B\right)=B\log \left(A\right)\\ & =& -\frac{n}{2}\log \left(2\pi {\sigma }^2\right)-\frac{1}{2{\sigma }^2}\sum _{k=1}^n(x_k-\mu {)}^2\cdots \log \left(e\right)=1\end{array}$$ 極値を考えるために変数である${\sigma }^2$で微分する(スコア凾数)． $$\begin{array}{r}\\ \frac{\mathrm{d}}{\mathrm{d}{\sigma }^2}l({\sigma }^2;x_1,\cdots ,x_n,\mu )& =& \frac{\mathrm{d}}{\mathrm{d}{\sigma }^2}\{-\frac{n}{2}\log \left(2\pi {\sigma }^2\right)-\frac{1}{2{\sigma }^2}\sum _{k=1}^n(x_k-\mu {)}^2\}\\ & =& -\frac{n}{2}\frac{\mathrm{d}}{\mathrm{d}{\sigma }^2}\log \left(2\pi {\sigma }^2\right)-\frac{1}{2}\{\sum _{k=1}^n(x_k-\mu {)}^2\}\frac{\mathrm{d}}{\mathrm{d}{\sigma }^2}\frac{1}{{\sigma }^2}\\ & & \cdots \frac{\mathrm{d}}{\mathrm{d}x}\{f(x)+g(x)\}=\frac{\mathrm{d}}{\mathrm{d}x}f(x)+\frac{\mathrm{d}}{\mathrm{d}x}g(x),\frac{\mathrm{d}}{\mathrm{d}x}cf(x)=c\frac{\mathrm{d}}{\mathrm{d}x}f(x)\\ & =& -\frac{n}{2}\frac{\mathrm{d}}{\mathrm{d}u}\log \left(u\right)\frac{\mathrm{d}u}{\mathrm{d}{\sigma }^2}-\frac{1}{2}\{\sum _{k=1}^n(x_k-\mu {)}^2\}\frac{\mathrm{d}}{\mathrm{d}v}{v}^{-1}\frac{\mathrm{d}v}{\mathrm{d}{\sigma }^2}\\ & & \cdots u=2\pi {\sigma }^2,\frac{\mathrm{d}u}{\mathrm{d}{\sigma }^2}=2\pi ,v={\sigma }^2,\frac{\mathrm{d}v}{\mathrm{d}{\sigma }^2}=1\\ & =& -\frac{n}{2}\frac{1}{u}2\pi -\frac{1}{2}\{\sum _{k=1}^n(x_k-\mu {)}^2\}(-v^{-2})\cdot 1\\ & =& -\frac{n}{2}\frac{1}{2\pi {\sigma }^2}2\pi -\frac{1}{2}\{\sum _{k=1}^n(x_k-\mu {)}^2\}\{-{\left({\sigma }^2\right)}^{-2}\}\cdots u=2\pi {\sigma }^2,v={\sigma }^2\\ & =& -\frac{n}{2}\frac{1}{{\sigma }^2}-\frac{1}{2}\{\sum _{k=1}^n(x_k-\mu {)}^2\}\left\{\frac{-1}{{\left({\sigma }^2\right)}^2}\right\}\\ & =& -\frac{n}{2{\sigma }^2}+\frac{1}{2{\sigma }^4}\{\sum _{k=1}^n(x_k-\mu {)}^2\}\cdots {\left(A^B\right)}^C=A^{BC}\end{array}$$ この式が0となる${\sigma }^2(={\hat{\sigma }}^2)$を求める(極値である)． $$\begin{array}{rcl}\frac{\mathrm{d}}{\mathrm{d}{\sigma }^2}l({\sigma }^2;x_1,\cdots ,x_n,\mu )=-\frac{n}{2{\hat{\sigma }}^2}+\frac{1}{2{\sigma }^4}\{\sum _{k=1}^n(x_k-\mu {)}^2\}& =& 0\\ \frac{1}{2{\hat{\sigma }}^2}[-n+\frac{1}{{\hat{\sigma }}^2}\{\sum _{k=1}^n(x_k-\mu {)}^2\}]& =& 0\\ -n+\frac{1}{{\hat{\sigma }}^2}\{\sum _{k=1}^n(x_k-\mu {)}^2\}& =& 0\cdots \frac{1}{2{\hat{\sigma }}^2}\mathrm{は}{\hat{\sigma }}^2\mathrm{が}\mathrm{有}\mathrm{限}\mathrm{な}\mathrm{ら}0\mathrm{に}\mathrm{な}\mathrm{ら}\mathrm{な}\mathrm{い}\mathrm{の}\mathrm{で}0\mathrm{に}\mathrm{な}\mathrm{る}\mathrm{の}\mathrm{は}\left[\right]\mathrm{の}\mathrm{中}\mathrm{が}0\mathrm{の}\mathrm{時}\\ \frac{1}{{\hat{\sigma }}^2}\{\sum _{k=1}^n(x_k-\mu {)}^2\}& =& n\\ \frac{1}{{\hat{\sigma }}^2}& =& \frac{n}{\sum _{k=1}^n(x_k-\mu {)}^2}\\ {\hat{\sigma }}^2& =& \frac{1}{n}\sum _{k=1}^n(x_k-\mu {)}^2\cdots \mathrm{両}\mathrm{辺}\mathrm{と}\mathrm{も}\mathrm{逆}\mathrm{数}\mathrm{を}\mathrm{と}\mathrm{っ}\mathrm{た}．\end{array}$$

$\mu$が未知の場合の${\sigma }^2$の最尤推定

対数尤度凾数を用意する(上と同じ)． $$\begin{array}{r}\\ l(\mu ,{\sigma }^2;x_1,\cdots ,x_n)& =& -\frac{n}{2}\log \left(2\pi {\sigma }^2\right)-\frac{1}{2{\sigma }^2}\sum _{k=1}^n(x_k-\mu {)}^2\end{array}$$ 極値を考えるために変数である$\mu ,{\sigma }^2$でそれぞれ偏微分する(スコア凾数)． $$\begin{array}{r}\\ \frac{\partial }{\partial \mu }l(\mu ,{\sigma }^2;x_1,\cdots ,x_n)& =& \frac{\partial }{\partial \mu }\{-\frac{n}{2}\log \left(2\pi {\sigma }^2\right)-\frac{1}{2{\sigma }^2}\sum _{k=1}^n(x_k-\mu {)}^2\}\\ & =& \frac{\partial }{\partial \mu }\{-\frac{n}{2}\log \left(2\pi {\sigma }^2\right)\}+\frac{\partial }{\partial \mu }\{-\frac{1}{2{\sigma }^2}\sum _{k=1}^n(x_k-\mu {)}^2\}\\ & & \cdots \frac{\partial }{\partial x}\{f(x,y)+g(x,y)\}=\frac{\partial }{\partial x}f(x,y)+\frac{\partial }{\partial x}g(x,y)\\ & =& -\frac{1}{2{\sigma }^2}\frac{\mathrm{d}}{\mathrm{d}\mu }\sum _{k=1}^n(x_k-\mu {)}^2\cdots \frac{\mathrm{d}}{\mathrm{d}x}c=0,\frac{\mathrm{d}}{\mathrm{d}x}cf(x)=c\frac{\mathrm{d}}{\mathrm{d}x}f(x)\\ & =& -\frac{1}{2{\sigma }^2}\sum _{k=1}^n\frac{\mathrm{d}}{\mathrm{d}\mu }(x_k-\mu {)}^2\\ & & \cdots \frac{\mathrm{d}}{\mathrm{d}x}\sum _{k=1}^{n}f(x_k)=\frac{\mathrm{d}}{\mathrm{d}x}\{f(x_1)+\cdots +f(x_n)\}=\frac{\mathrm{d}}{\mathrm{d}x}f(x_1)+\cdots +\frac{\mathrm{d}}{\mathrm{d}x}f(x_n)=\sum _{k=1}^n\frac{\mathrm{d}}{\mathrm{d}x}f(x_k)\\ & =& -\frac{1}{2{\sigma }^2}\sum _{k=1}^n\frac{\mathrm{d}}{\mathrm{d}u}{u}^2\frac{\mathrm{d}u}{\mathrm{d}\mu }\cdots u=x_k-\mu ,\frac{\mathrm{d}u}{\mathrm{d}\mu }=-1\\ & =& -\frac{1}{2{\sigma }^2}\sum _{k=1}^{n}2u\cdot -1\\ & =& -\frac{1}{2{\sigma }^2}\sum _{k=1}^n-2(x_k-\mu )\cdots u=x_k-\mu \\ & =& -\frac{1}{2{\sigma }^2}(-2)\sum _{k=1}^n(x_k-\mu )\\ & =& \frac{1}{{\sigma }^2}\sum _{k=1}^n(x_k-\mu )\\ \frac{\partial }{\partial {\sigma }^2}l(\mu ,{\sigma }^2;x_1,\cdots ,x_n)& =& \frac{\partial }{\partial {\sigma }^2}\{-\frac{n}{2}\log \left(2\pi {\sigma }^2\right)-\frac{1}{2{\sigma }^2}\sum _{k=1}^n(x_k-\mu {)}^2\}\\ & =& -\frac{n}{2{\sigma }^2}+\frac{1}{2{\sigma }^4}\{\sum _{k=1}^n(x_k-\mu {)}^2\}\cdots l({\sigma }^2;x_1,\cdots ,x_n,\mu )\mathrm{の}\mathrm{時}\mathrm{と}\mathrm{同}\mathrm{様}\end{array}$$ これらの式が0となる連立方程式として$\mu (=\hat{\mu }),{\sigma }^2(={\tilde{\sigma }}^2)$を求める(極値である)． $$\begin{array}{r}\{\begin{array}{}\frac{1}{{\tilde{\sigma }}^2}\sum _{k=1}^n(x_k-\hat{\mu })& =& 0\\ -\frac{n}{2{\tilde{\sigma }}^2}+\frac{1}{2{\tilde{\sigma }}^4}\{\sum _{k=1}^n(x_k-\hat{\mu }{)}^2\}& =& 0\end{array}\end{array}$$ 第一式より $$\begin{array}{rcl}\frac{1}{{\tilde{\sigma }}^2}\sum _{k=1}^n(x_k-\hat{\mu })& =& 0\\ \sum _{k=1}^n(x_k-\hat{\mu })& =& 0\cdots \frac{1}{{\tilde{\sigma }}^2}\mathrm{は}{\tilde{\sigma }}^2\mathrm{が}\mathrm{有}\mathrm{限}\mathrm{な}\mathrm{ら}0\mathrm{に}\mathrm{な}\mathrm{ら}\mathrm{な}\mathrm{い}\mathrm{の}\mathrm{で}\sum \mathrm{以}\mathrm{降}\mathrm{が}0\\ \sum _{k=1}^n{x}_k-\sum _{k=1}^n\hat{\mu }& =& 0\cdots \sum (A-B)=\sum A-\sum B\\ \sum _{k=1}^n{x}_k-n\hat{\mu }& =& 0\cdots \sum _{k=1}^{n}c=nc(c:\mathrm{定}\mathrm{数})\\ -n\hat{\mu }& =& -\sum _{k=1}^n{x}_k\\ \hat{\mu }& =& \frac{1}{n}\sum _{k=1}^n{x}_k\\ & =& \overline{x}\cdots \mathrm{標}\mathrm{本}\mathrm{平}\mathrm{均}\overline{x}=\frac{1}{n}\sum _{k=1}^n{x}_k\end{array}$$ 第二式の$\hat{\mu }$にこれを代入して${\tilde{\sigma }}^2$の最尤推定量を得る． $$\begin{array}{rcl}-\frac{n}{2{\tilde{\sigma }}^2}+\frac{1}{2{\tilde{\sigma }}^4}\{\sum _{k=1}^n(x_k-\hat{\mu }{)}^2\}& =& 0\\ -\frac{n}{2{\tilde{\sigma }}^2}+\frac{1}{2{\tilde{\sigma }}^4}\{\sum _{k=1}^n(x_k-\overline{x}{)}^2\}& =& 0\cdots \hat{\mu }=\overline{x}\\ \frac{1}{2{\tilde{\sigma }}^2}[-n+\frac{1}{{\tilde{\sigma }}^2}\{\sum _{k=1}^n(x_k-\overline{x}{)}^2\}]& =& 0\\ -n+\frac{1}{{\tilde{\sigma }}^2}\{\sum _{k=1}^n(x_k-\overline{x}{)}^2\}& =& 0\cdots \frac{1}{2{\tilde{\sigma }}^2}\mathrm{は}{\tilde{\sigma }}^2\mathrm{が}\mathrm{有}\mathrm{限}\mathrm{な}\mathrm{ら}0\mathrm{に}\mathrm{な}\mathrm{ら}\mathrm{な}\mathrm{い}\mathrm{の}\mathrm{で}0\mathrm{に}\mathrm{な}\mathrm{る}\mathrm{の}\mathrm{は}\left[\right]\mathrm{の}\mathrm{中}\mathrm{が}0\mathrm{の}\mathrm{時}\\ \frac{1}{{\tilde{\sigma }}^2}\{\sum _{k=1}^n(x_k-\overline{x}{)}^2\}& =& n\\ \frac{1}{{\tilde{\sigma }}^2}& =& \frac{n}{\sum _{k=1}^n(x_k-\overline{x}{)}^2}\\ {\tilde{\sigma }}^2& =& \frac{1}{n}\sum _{k=1}^n(x_k-\overline{x}{)}^2\\ & =& s^2\cdots \mathrm{標}\mathrm{本}\mathrm{分}\mathrm{散}{s}^2=\frac{1}{n}\sum _{k=1}^n(X_k-\bar{X}{)}^2\\ & =& \frac{n-1}{n}{\hat{\sigma }}^2\cdots \mathrm{不}\mathrm{偏}\mathrm{分}\mathrm{散}{\hat{\sigma }}^2=\frac{1}{n-1}\sum _{k=1}^n(X_k-\bar{X}{)}^2\\ & & \cdots \mathrm{こ}\mathrm{れ}\mathrm{は}\mathrm{標}\mathrm{本}\mathrm{分}\mathrm{散}\mathrm{の}\mathrm{期}\mathrm{待}\mathrm{値}\mathrm{で}\mathrm{も}\mathrm{あ}\mathrm{る}．\end{array}$$

XY平面上の線分同士の交点

線分同士の交点

$XY\mathrm{平}\mathrm{面}\mathrm{上}\mathrm{の}\mathrm{線}\mathrm{分}\overline{AB},\overline{CD}$の$\mathrm{交}\mathrm{点}G$

$$\begin{array}{rcl}\overrightarrow{u}& =& \overrightarrow{AB}=(u_x,u_y)=(x_b-x_a,y_b-y_a)\cdots \mathrm{始}\mathrm{点}A(x_a,y_a),\mathrm{終}\mathrm{点}B(x_b,y_b)\\ \overrightarrow{v}& =& \overrightarrow{CD}=(v_x,v_y)=(x_d-x_c,y_d-y_c)\cdots \mathrm{始}\mathrm{点}C(x_c,y_c),\mathrm{終}\mathrm{点}D(x_d,y_d)\\ \overrightarrow{w}& =& \overrightarrow{AC}=(w_x,w_y)=(x_c-x_a,y_c-y_a)\cdots \mathrm{始}\mathrm{点}A,\mathrm{終}\mathrm{点}B\\ \overrightarrow{u}\times \overrightarrow{v}& =& u_x{v}_y-v_x{u}_y=(x_b-x_a)(y_d-yc)-(x_d-x_c)(y_b-y_a)\\ \sin \left(\theta \right)& =& \frac{\overrightarrow{u}\times \overrightarrow{v}}{\left|\overrightarrow{u}\right|\left|\overrightarrow{v}\right|}\\ & & \cdots \overrightarrow{\alpha }\times \overrightarrow{\beta }=XY\mathrm{平}\mathrm{面}\mathrm{に}\mathrm{垂}\mathrm{直}\mathrm{な}\mathrm{成}\mathrm{分}=\left|\overrightarrow{\alpha }\right|\left|\overrightarrow{\beta }\right|\sin \left(x\right)\\ & & \cdots \mathrm{本}\mathrm{来}\mathrm{外}\mathrm{積}\mathrm{は}\mathrm{演}\mathrm{算}\mathrm{に}\mathrm{用}\mathrm{い}\mathrm{た}2\mathrm{つ}\mathrm{の}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}\mathrm{に}\mathrm{垂}\mathrm{直}\mathrm{な}\mathrm{軸}\mathrm{の}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}\mathrm{が}\mathrm{演}\mathrm{算}\mathrm{結}\mathrm{果}\mathrm{で}\mathrm{あ}\mathrm{り}\\ & & \cdots \sin (\theta )\mathrm{を}\mathrm{求}\mathrm{め}\mathrm{る}\mathrm{に}\mathrm{は}\mathrm{長}\mathrm{さ}(\mathrm{絶}\mathrm{対}\mathrm{値})\mathrm{を}\mathrm{取}\mathrm{っ}\mathrm{て}\mathrm{計}\mathrm{算}\mathrm{す}\mathrm{る}\mathrm{こ}\mathrm{と}\mathrm{に}\mathrm{な}\mathrm{る}\mathrm{の}\mathrm{だ}\mathrm{が}，\\ & & \cdots \mathrm{こ}\mathrm{こ}\mathrm{で}\mathrm{は}XY\mathrm{平}\mathrm{面}\mathrm{に}\mathrm{対}\mathrm{し}\mathrm{て}\mathrm{垂}\mathrm{直}\mathrm{な}\mathrm{成}\mathrm{分}\mathrm{の}\mathrm{み}\mathrm{の}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}\mathrm{な}\mathrm{の}\mathrm{で}，\\ & & \cdots \mathrm{そ}\mathrm{の}XY\mathrm{平}\mathrm{面}\mathrm{に}\mathrm{対}\mathrm{し}\mathrm{て}\mathrm{垂}\mathrm{直}\mathrm{な}\mathrm{成}\mathrm{分}\mathrm{を}\mathrm{ス}\mathrm{カ}\mathrm{ラ}\mathrm{ー}\mathrm{と}\mathrm{し}\mathrm{て}\mathrm{扱}\mathrm{う}\mathrm{こ}\mathrm{と}\mathrm{で}\mathrm{長}\mathrm{さ}(\mathrm{絶}\mathrm{対}\mathrm{値})\mathrm{と}\mathrm{し}\mathrm{て}\mathrm{扱}\mathrm{う}\mathrm{の}\mathrm{と}\mathrm{異}\mathrm{な}\mathrm{り}\\ & & \cdots \sin (\theta )<0\mathrm{が}\mathrm{扱}\mathrm{え}\mathrm{る}．\\ \sin (-\theta )& =& \frac{\overrightarrow{v}\times \overrightarrow{u}}{\left|\overrightarrow{v}\right|\left|\overrightarrow{u}\right|}\end{array}$$

$\mathrm{点}A\mathrm{か}\mathrm{ら}\mathrm{交}\mathrm{点}G\mathrm{ま}\mathrm{で}\mathrm{の}\mathrm{長}\mathrm{さ}\overline{AG}$と$\mathrm{線}\mathrm{分}\mathrm{の}\mathrm{長}\mathrm{さ}\overline{AB}$の比$\frac{\overline{AG}}{\overline{AB}}$

$$\begin{array}{rcl}\overline{AE}& =& \left|\overrightarrow{w}\right|\sin (\pi -\varphi )\\ & =& \left|\overrightarrow{w}\right|\sin \left(\varphi \right)\cdots \sin (\pi -x)=\sin \left(x\right)\\ & =& \left|\overrightarrow{w}\right|\frac{\overrightarrow{w}\times \overrightarrow{v}}{\left|\overrightarrow{w}\right|\left|\overrightarrow{v}\right|}\\ & & \cdots \overrightarrow{\alpha }\times \overrightarrow{\beta }=\left|\overrightarrow{\alpha }\right|\left|\overrightarrow{\beta }\right|\sin \left(x\right),\sin \left(x\right)=\frac{\overrightarrow{\alpha }\times \overrightarrow{\beta }}{\left|\overrightarrow{\alpha }\right|\left|\overrightarrow{\beta }\right|}\\ & & \cdots \mathrm{前}\mathrm{述}\mathrm{の}\mathrm{通}\mathrm{り}\mathrm{外}\mathrm{積}\mathrm{の}\mathrm{結}\mathrm{果}\mathrm{を}\mathrm{ス}\mathrm{カ}\mathrm{ラ}\mathrm{ー}\mathrm{と}\mathrm{し}\mathrm{て}\mathrm{扱}\mathrm{う}．\\ & =& \frac{\overrightarrow{w}\times \overrightarrow{v}}{\left|\overrightarrow{v}\right|}\\ \overline{AE}& =& \overline{AG}\sin (\pi -\theta )\\ & =& \overline{AG}\sin \left(\theta \right)\cdots \sin (\pi -x)=\sin \left(x\right)\\ \overline{AG}& =& \frac{\overline{AE}}{\sin \left(\theta \right)}\\ & =& \frac{\overrightarrow{w}\times \overrightarrow{v}}{\left|\overrightarrow{v}\right|}\frac{1}{\sin \left(\theta \right)}\\ & =& \frac{\overrightarrow{w}\times \overrightarrow{v}}{\left|\overrightarrow{v}\right|}\frac{\left|\overrightarrow{u}\right|\left|\overrightarrow{v}\right|}{\overrightarrow{u}\times \overrightarrow{v}}\\ & =& \frac{\overrightarrow{w}\times \overrightarrow{v}}{\overrightarrow{u}\times \overrightarrow{v}}\left|\overrightarrow{u}\right|\\ \frac{\overline{AG}}{\overline{AB}}& =& \frac{\overline{AG}}{\left|\overrightarrow{u}\right|}\\ & =& \frac{\overrightarrow{w}\times \overrightarrow{v}}{\overrightarrow{u}\times \overrightarrow{v}}\left|\overrightarrow{u}\right|\frac{1}{\left|\overrightarrow{u}\right|}\\ & =& \frac{\overrightarrow{w}\times \overrightarrow{v}}{\overrightarrow{u}\times \overrightarrow{v}}\\ & =& \frac{w_x{v}_y-v_x{w}_y}{u_x{v}_y-v_x{u}_y}\\ & =& \frac{(x_c-x_a)(y_d-y_c)-(x_d-x_c)(y_c-y_a)}{(x_b-x_a)(y_d-y_c)-(x_d-x_c)(y_b-y_a)}\end{array}$$

$\mathrm{点}C\mathrm{か}\mathrm{ら}\mathrm{交}\mathrm{点}G\mathrm{ま}\mathrm{で}\mathrm{の}\mathrm{長}\mathrm{さ}\overline{CG}$と$\mathrm{線}\mathrm{分}\mathrm{の}\mathrm{長}\mathrm{さ}\overline{CD}$の比$\frac{\overline{CG}}{\overline{CD}}$

$$\begin{array}{rcl}\overline{CF}& =& |-\overrightarrow{w}|\sin (\pi -\psi )\\ & =& |-\overrightarrow{w}|\sin \left(\psi \right)\cdots \sin (\pi -x)=\sin \left(x\right)\\ & =& |-\overrightarrow{w}|\frac{-\overrightarrow{w}\times \overrightarrow{u}}{|-\overrightarrow{w}|\left|\overrightarrow{u}\right|}\\ & & \cdots \overrightarrow{\alpha }\times \overrightarrow{\beta }=\left|\overrightarrow{\alpha }\right|\left|\overrightarrow{\beta }\right|\sin \left(x\right),\sin \left(x\right)=\frac{\overrightarrow{\alpha }\times \overrightarrow{\beta }}{\left|\overrightarrow{\alpha }\right|\left|\overrightarrow{\beta }\right|}\\ & & \cdots \mathrm{前}\mathrm{述}\mathrm{の}\mathrm{通}\mathrm{り}\mathrm{外}\mathrm{積}\mathrm{の}\mathrm{結}\mathrm{果}\mathrm{を}\mathrm{ス}\mathrm{カ}\mathrm{ラ}\mathrm{ー}\mathrm{と}\mathrm{し}\mathrm{て}\mathrm{扱}\mathrm{う}．\\ & =& \frac{-\overrightarrow{w}\times \overrightarrow{u}}{\left|\overrightarrow{u}\right|}\\ \overline{CF}& =& \overline{CG}\sin (-(\pi -\theta ))\\ & =& \overline{CG}\sin (-\theta )\cdots \sin (\pi -x)=\sin \left(x\right)\\ \overline{CG}& =& \frac{\overline{CF}}{\sin (-\theta )}\\ & =& \frac{-\overrightarrow{w}\times \overrightarrow{u}}{\left|\overrightarrow{u}\right|}\frac{1}{\sin (-\theta )}\\ & =& \frac{-\overrightarrow{w}\times \overrightarrow{u}}{\left|\overrightarrow{u}\right|}\frac{\left|\overrightarrow{v}\right|\left|\overrightarrow{u}\right|}{\overrightarrow{v}\times \overrightarrow{u}}\\ & =& \frac{-\overrightarrow{w}\times \overrightarrow{u}}{\overrightarrow{v}\times \overrightarrow{u}}\left|\overrightarrow{v}\right|\\ \frac{\overline{CG}}{\overline{CD}}& =& \frac{\overline{CG}}{\left|\overrightarrow{v}\right|}\\ & =& \frac{-\overrightarrow{w}\times \overrightarrow{u}}{\overrightarrow{v}\times \overrightarrow{u}}\left|\overrightarrow{v}\right|\frac{1}{\left|\overrightarrow{v}\right|}\\ & =& \frac{-\overrightarrow{w}\times \overrightarrow{u}}{\overrightarrow{v}\times \overrightarrow{u}}\\ & =& \frac{-\overrightarrow{w}\times \overrightarrow{u}}{-(\overrightarrow{u}\times \overrightarrow{v})}\cdots A\times B=-(B\times A)\\ & =& \frac{\overrightarrow{w}\times \overrightarrow{u}}{\overrightarrow{u}\times \overrightarrow{v}}\\ & =& \frac{w_x{u}_y-u_x{w}_y}{u_x{v}_y-v_x{u}_y}\\ & =& \frac{(x_c-x_a)(y_b-y_a)-(x_b-x_a)(y_c-y_a)}{(x_b-x_a)(y_d-y_c)-(x_d-x_c)(y_b-y_a)}\end{array}$$

交点の有無について

$\frac{\overline{CG}}{\overline{CD}}$と$\frac{\overline{AG}}{\overline{AB}}$は共に 分母に$\overrightarrow{u}\times \overrightarrow{v}$があるが， これが$0$というのは$\sin \left(\theta \right)=0$ということであり，線分同士がなす角$\theta$が$0\mathrm{か}\pi$であり，互いに$\overline{AB},\overline{CD}$が平行ということなので$\mathrm{交}\mathrm{点}G$を持たない． $0$での割り算を発生させないためこれは例外として$\frac{\overline{CG}}{\overline{CD}}$と$\frac{\overline{AG}}{\overline{AB}}$を求めるより先に判定する必要がある．
その後，$\frac{\overline{CG}}{\overline{CD}}$と$\frac{\overline{AG}}{\overline{AB}}$を求め，共に$0$より大きく$1$以下の値の時，$\mathrm{交}\mathrm{点}G$を持つことになる． それ以外の場合は$\mathrm{交}\mathrm{点}G$が線分の外側に存在することになる．

単回帰における残差平方和の期待値

単回帰における残差平方和の期待値

誤差の平方和

$$\begin{array}{rcl}\sum _{i=1}^n{ϵ}_i^2& =& \sum _{i=1}^n{(y_i-\alpha -\beta x_i)}^2\cdots y_i=\alpha +\beta x_i+{ϵ}_i,\mathrm{誤}\mathrm{差}:{ϵ}_i\stackrel{iid}{\sim }N(0,{\sigma }^2)\mathrm{互}\mathrm{い}\mathrm{に}\mathrm{独}\mathrm{立}\mathrm{な}N(0,{\sigma }^2)\mathrm{分}\mathrm{布}\mathrm{に}\mathrm{従}\mathrm{う}\mathrm{と}\mathrm{仮}\mathrm{定}\mathrm{す}\mathrm{る}.\\ & =& \sum _{i=1}^n{(y_i-\alpha -\beta x_i+\overline{y}-\hat{\alpha }-\hat{\beta }\overline{x}+\hat{\beta }{x}_i-\hat{\beta }{x}_i+\beta \overline{x}-\beta \overline{x})}^2\cdots \overline{y}-\hat{\alpha }-\hat{\beta }\overline{x}=0\\ & =& \sum _{i=1}^n{(y_i-\hat{\alpha }-\hat{\beta }{x}_i+\overline{y}-\alpha -\beta \overline{x}+\hat{\beta }{x}_i-\hat{\beta }\overline{x}-\beta x_i+\beta \overline{x})}^2\\ & =& \sum _{i=1}^n{[(y_i-\hat{\alpha }-\hat{\beta }{x}_i)+(\overline{y}-\alpha -\beta \overline{x})+(\hat{\beta }{x}_i-\hat{\beta }\overline{x}-\beta x_i+\beta \overline{x})]}^2\\ & =& \sum _{i=1}^n{[(y_i-\hat{\alpha }-\hat{\beta }{x}_i)+(\overline{y}-\alpha -\beta \overline{x})+\left\{(\hat{\beta }-\beta )(x_i-\overline{x})\right\}]}^2\\ & =& \sum _{i=1}^n[{(y_i-\hat{\alpha }-\hat{\beta }{x}_i)}^2+{(\overline{y}-\alpha -\beta \overline{x})}^2+{\left\{(\hat{\beta }-\beta )(x_i-\overline{x})\right\}}^2+2(y_i-\hat{\alpha }-\hat{\beta }{x}_i)(\overline{y}-\alpha -\beta \overline{x})+2(y_i-\hat{\alpha }-\hat{\beta }{x}_i)(\hat{\beta }-\beta )(x_i-\overline{x})+2(\overline{y}-\alpha -\beta \overline{x})(\hat{\beta }-\beta )(x_i-\overline{x})]\\ & & \cdots (A+B+C{)}^2=A^2+B^2+C^2+2AB+2AC+2BC\\ & =& \sum _{i=1}^n{(y_i-\hat{\alpha }-\hat{\beta }{x}_i)}^2+\sum _{i=1}^n{(\overline{y}-\alpha -\beta \overline{x})}^2+\sum _{i=1}^n{(\hat{\beta }-\beta )}^2{(x_i-\overline{x})}^2\\ & & +2\sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)(\overline{y}-\alpha -\beta \overline{x})+2\sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)(\hat{\beta }-\beta )(x_i-\overline{x})+2\sum _{i=1}^n(\overline{y}-\alpha -\beta \overline{x})(\hat{\beta }-\beta )(x_i-\overline{x})\\ & =& \sum _{i=1}^n{(y_i-\hat{\alpha }-\hat{\beta }{x}_i)}^2+\sum _{i=1}^n{(\overline{y}-\alpha -\beta \overline{x})}^2+{(\hat{\beta }-\beta )}^2\sum _{i=1}^n{(x_i-\overline{x})}^2+2\cdot 0+2\cdot 0+2\cdot 0\\ & & \cdots \sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)(\overline{y}-\alpha -\beta \overline{x})=0(\mathrm{後}\mathrm{述})\\ & & \cdots \sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)(\hat{\beta }-\beta )(x_i-\overline{x})=0(\mathrm{後}\mathrm{述})\\ & & \cdots \sum _{i=1}^n(\overline{y}-\alpha -\beta \overline{x})(\hat{\beta }-\beta )(x_i-\overline{x})＝0(\mathrm{後}\mathrm{述})\\ & =& \sum _{i=1}^n{(y_i-\widehat{y_i})}^2+n{(\overline{y}-\alpha -\beta \overline{x})}^2+{(\hat{\beta }-\beta )}^2{S}_{xx}\cdots \widehat{y_i}=\hat{\alpha }+\hat{\beta }{x}_i,S_{xx}=\sum _{i=1}^n{(x_i-\overline{x})}^2\\ & =& \sum _{i=1}^n{e}_i^2+n{(\overline{y}-\alpha -\beta \overline{x})}^2+{(\hat{\beta }-\beta )}^2{S}_{xx}\cdots e_i=y_i-\widehat{y_i}\end{array}$$

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残差平方和$S_e$

前述の式で$\sum _{i=1}^n{ϵ}_i^2$と$\sum _{i=1}^n{e}_i^2$を入れ替えて(互に移項して)以下の式を得る． $$\begin{array}{rcl}\sum _{i=1}^n{e}_i^2& =& \sum _{i=1}^n{(y_i-\widehat{y_i})}^2\cdots \mathrm{残}\mathrm{差}\mathrm{平}\mathrm{方}\mathrm{和}(sumofsquaresofresiduals;S_e)\\ & =& \sum _{i=1}^n{ϵ}_i^2-n{(\overline{y}-\alpha -\beta \overline{x})}^2-{(\hat{\beta }-\beta )}^2{S}_{xx}\end{array}$$

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残差平方和の期待値$\mathrm{E}\left[S_e\right]$

$$\begin{array}{rcl}\mathrm{E}\left[\sum _{i=1}^n{e}_i^2\right]& =& \mathrm{E}[\sum _{i=1}^n{ϵ}_i^2-n{(\overline{y}-\alpha -\beta \overline{x})}^2-{(\hat{\beta }-\beta )}^2{S}_{xx}]\\ & =& \mathrm{E}\left[\sum _{i=1}^n{ϵ}_i^2\right]-\mathrm{E}\left[n{(\overline{y}-\alpha -\beta \overline{x})}^2\right]-\mathrm{E}\left[{(\hat{\beta }-\beta )}^2{S}_{xx}\right]\cdots \mathrm{E}[X+Y]=\mathrm{E}\left[X\right]+\mathrm{E}\left[Y\right]\\ & =& \sum _{i=1}^n\mathrm{E}\left[{ϵ}_i^2\right]-\mathrm{E}\left[n{(\overline{y}-\alpha -\beta \overline{x})}^2\right]-\mathrm{E}\left[{(\hat{\beta }-\beta )}^2{S}_{xx}\right]\\ & & \cdots \mathrm{E}\left[\sum _{i=1}^n{X}_i\right]=\mathrm{E}[X_1+\cdots +X_n]=\mathrm{E}\left[X_1\right]+\cdots +\mathrm{E}\left[X_n\right]=\sum _{i=1}^n\mathrm{E}\left[X_i\right]\\ & =& \sum _{i=1}^n(\mathrm{V}\left[{ϵ}_i\right]+\mathrm{E}{\left[{ϵ}_i\right]}^2)-\mathrm{E}\left[n{(\overline{y}-\alpha -\beta \overline{x})}^2\right]-\mathrm{E}\left[{(\hat{\beta }-\beta )}^2{S}_{xx}\right]\\ & & \cdots \mathrm{V}\left[X\right]=\mathrm{E}\left[{(X-\mathrm{E}\left[X\right])}^2\right]=\mathrm{E}\left[X^2\right]-\mathrm{E}{\left[X\right]}^2,\mathrm{E}\left[X^2\right]=\mathrm{V}\left[X\right]+\mathrm{E}{\left[X\right]}^2\\ & =& \sum _{i=1}^n({\sigma }^2+0^2)-\mathrm{E}\left[n{(\overline{y}-\alpha -\beta \overline{x})}^2\right]-\mathrm{E}\left[{(\hat{\beta }-\beta )}^2{S}_{xx}\right]\cdots {ϵ}_i\stackrel{iid}{\sim }N(0,{\sigma }^2),\mathrm{E}\left[{ϵ}_i\right]=0,\mathrm{V}\left[{ϵ}_i\right]={\sigma }^2\\ & =& \sum _{i=1}^n{\sigma }^2-\mathrm{E}\left[n{(\overline{y}-\alpha -\beta \overline{x})}^2\right]-\mathrm{E}\left[{(\hat{\beta }-\beta )}^2{S}_{xx}\right]\\ & =& {\sigma }^2\sum _{i=1}^{n}1-\mathrm{E}\left[n{(\overline{y}-\alpha -\beta \overline{x})}^2\right]-\mathrm{E}\left[{(\hat{\beta }-\beta )}^2{S}_{xx}\right]\\ & =& n{\sigma }^2-n\mathrm{E}\left[{(\overline{y}-\alpha -\beta \overline{x})}^2\right]-S_{xx}\mathrm{E}\left[{(\hat{\beta }-\beta )}^2\right]\cdots \mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]\\ & =& n{\sigma }^2-n\mathrm{V}\left[\overline{y}\right]-S_{xx}\mathrm{V}\left[\hat{\beta }\right]\cdots \mathrm{V}[X+t]=\mathrm{V}\left[X\right](t:\mathrm{定}\mathrm{数})\\ & =& n{\sigma }^2-n\frac{{\sigma }^2}{n}-S_{xx}\mathrm{V}\left[\frac{S_{xy}}{S_{xx}}\right]\cdots \mathrm{V}\left[\overline{y}\right]=\frac{{\sigma }^2}{n},\hat{\beta }=\frac{S_{xy}}{S_{xx}}\\ & =& n{\sigma }^2-n\frac{{\sigma }^2}{n}-S_{xx}\frac{1}{S_{xx}^2}\mathrm{V}\left[S_{xy}\right]\cdots \mathrm{V}\left[cX\right]=c^2\mathrm{V}\left[X\right](c:\mathrm{定}\mathrm{数})\\ & =& n{\sigma }^2-{\sigma }^2-\frac{1}{S_{xx}}{\sigma }^2{S}_{xx}\cdots \mathrm{V}\left[S_{xy}\right]={\sigma }^2{S}_{xx}\\ & =& n{\sigma }^2-{\sigma }^2-{\sigma }^2\\ & =& (n-2){\sigma }^2\cdots \mathrm{残}\mathrm{差}\mathrm{平}\mathrm{方}\mathrm{和}(S_e)\mathrm{の}\mathrm{期}\mathrm{待}\mathrm{値}\end{array}$$ よって$(n-2)$で残差平方和を割っておけば${\sigma }^2$が得られ不偏推定量となる． $$\begin{array}{rcl}\mathrm{E}\left[\sum _{i=1}^n{e}_i^2\right]& =& (n-2){\sigma }^2\\ \frac{1}{n-2}\mathrm{E}\left[\sum _{i=1}^n{e}_i^2\right]& =& {\sigma }^2\\ \mathrm{E}\left[\frac{1}{(n-2)}\sum _{i=1}^n{e}_i^2\right]& =& {\sigma }^2\cdots c\mathrm{E}\left[X\right]=\mathrm{E}\left[cX\right]\\ \mathrm{E}\left[s^2\right]& =& {\sigma }^2\cdots s^2=\frac{1}{(n-2)}\sum _{i=1}^n{e}_i^2\mathrm{は}{\sigma }^2\mathrm{の}\mathrm{不}\mathrm{偏}\mathrm{推}\mathrm{定}\mathrm{量}\end{array}$$

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標準化残差$e_{is}$とその分布

$$\begin{array}{r}\\ e_{is}& \equiv & \frac{e_i}{s}\cdots \mathrm{標}\mathrm{準}\mathrm{化}\mathrm{残}\mathrm{差}(standardizedresiduals),s\equiv \sqrt{\frac{1}{(n-2)}\sum _{i=1}^n{e}_i^2}\\ e_i& =& y_i-\widehat{y_i}=(\alpha +\beta x_i+{ϵ}_i)-(\hat{\alpha }+\hat{\beta }{x}_i)\\ & =& (\alpha -\hat{\alpha })+(\beta -\hat{\beta })x_i+{ϵ}_i\end{array}$$ $n$が十分大きい時$\hat{\alpha },\hat{\beta }$は$\alpha ,\beta$に確率収束する( $\hat{\alpha },\hat{\beta }$は不偏推定量(期待値は$\alpha ,\beta$) )．よって $$\begin{array}{rcl}{e}_i& \sim & {ϵ}_i\cdots (\alpha -\hat{\alpha }),(\beta -\hat{\beta })\mathrm{は}\mathrm{ほ}\mathrm{ぼ}0\mathrm{に}\mathrm{な}\mathrm{り}\mathrm{第}1,2\mathrm{項}\mathrm{は}\mathrm{無}\mathrm{視}\mathrm{で}\mathrm{き}\mathrm{る}\mathrm{よ}\mathrm{う}\mathrm{に}\mathrm{な}\mathrm{る}．\end{array}$$ となる．また， $$\begin{array}{rcl}{e}_{is}& =& \frac{e_i}{s}\end{array}$$ は， $$\begin{array}{rcl}{ϵ}_{is}& \equiv & \frac{{ϵ}_i}{\sigma }\cdots n\mathrm{が}\mathrm{十}\mathrm{分}\mathrm{大}\mathrm{き}\mathrm{い}\mathrm{時},s\mathrm{は}\sigma \mathrm{に}\mathrm{確}\mathrm{率}\mathrm{収}\mathrm{束}\mathrm{す}\mathrm{る}(s^2\mathrm{は}\mathrm{不}\mathrm{偏}\mathrm{推}\mathrm{定}\mathrm{量}(\mathrm{期}\mathrm{待}\mathrm{値}\mathrm{は}{\sigma }^2))．\end{array}$$ と同じ分布に従うとみなせる．
${ϵ}_i\sim N(0,{\sigma }^2)$に従うので${ϵ}_{is}$は$N(0,1^2)$に従うと考えられる．

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誤差の平方和の計算途中の式について

$$\begin{array}{rcl}\sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)(\overline{y}-\alpha -\beta \overline{x})& =& (\overline{y}-\alpha -\beta \overline{x})\sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)\\ & =& (\overline{y}-\alpha -\beta \overline{x})(\sum _{i=1}^n{y}_i-\hat{\alpha }\sum _{i=1}^{n}1-\hat{\beta }\sum _{i=1}^n{x}_i)\\ & =& (\overline{y}-\alpha -\beta \overline{x})(n\overline{y}-n\hat{\alpha }-n\hat{\beta }\overline{x})\\ & =& (\overline{y}-\alpha -\beta \overline{x})n(\overline{y}-\hat{\alpha }-\hat{\beta }\overline{x})\\ & =& (\overline{y}-\alpha -\beta \overline{x})n\cdot 0\cdots \overline{y}-\hat{\alpha }-\hat{\beta }\overline{x}=0\\ & =& 0\\ \sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)(\hat{\beta }-\beta )(x_i-\overline{x})& =& (\hat{\beta }-\beta )\sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)(x_i-\overline{x})\\ & =& (\hat{\beta }-\beta )\sum _{i=1}^n\{(y_i-\hat{\alpha }-\hat{\beta }{x}_i)x_i-(y_i-\hat{\alpha }-\hat{\beta }{x}_i)\overline{x}\}\\ & =& (\hat{\beta }-\beta )\{\sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)x_i-\sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)\overline{x}\}\\ & =& (\hat{\beta }-\beta )\{\sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)x_i-\overline{x}\sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)\}\\ & =& (\hat{\beta }-\beta )\{\sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)x_i-\overline{x}(\sum _{i=1}^n{y}_i-\sum _{i=1}^n\hat{\alpha }-\sum _{i=1}^n\hat{\beta }{x}_i)\}\\ & =& (\hat{\beta }-\beta )\{\sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)x_i-\overline{x}(n\overline{y}-n\hat{\alpha }-\hat{\beta }n\overline{x})\}\\ & =& (\hat{\beta }-\beta )\{\sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)x_i-\overline{x}n(\overline{y}-\hat{\alpha }-\hat{\beta }\overline{x})\}\\ & =& (\hat{\beta }-\beta )\{\sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)x_i-\overline{x}n\cdot 0\}\\ & =& (\hat{\beta }-\beta )\{\sum _{i=1}^n(y_i-\hat{\alpha }-\hat{\beta }{x}_i)x_i-0\}\\ & =& (\hat{\beta }-\beta )\sum _{i=1}^n(y_i{x}_i-\hat{\alpha }{x}_i-\hat{\beta }{x}_i^2)\\ & =& (\hat{\beta }-\beta )\{\sum _{i=1}^n{y}_i{x}_i-\sum _{i=1}^n\hat{\alpha }{x}_i-\sum _{i=1}^n\hat{\beta }{x}_i^2\}\\ & =& (\hat{\beta }-\beta )\{\sum _{i=1}^n{y}_i{x}_i-\hat{\alpha }\sum _{i=1}^n{x}_i-\hat{\beta }\sum _{i=1}^n{x}_i^2\}\\ & =& (\hat{\beta }-\beta )\{\sum _{i=1}^n{y}_i{x}_i-\hat{\alpha }n\overline{x}-\hat{\beta }\sum _{i=1}^n{x}_i^2\}\cdots \sum _{i=1}^n{y}_i{x}_i-\hat{\alpha }n\overline{x}-\hat{\beta }\sum _{i=1}^n{x}_i^2=0(\mathrm{正}\mathrm{規}\mathrm{方}\mathrm{程}\mathrm{式}\mathrm{の}\mathrm{一}\mathrm{つ}．\hat{\alpha },\hat{\beta }\mathrm{は}\mathrm{こ}\mathrm{の}\mathrm{式}\mathrm{を}\mathrm{満}\mathrm{た}\mathrm{す}\mathrm{た}\mathrm{め}\mathrm{に}\mathrm{調}\mathrm{整}\mathrm{し}\mathrm{た}\mathrm{値})\\ & =& (\hat{\beta }-\beta )\cdot 0\\ & =& 0\\ \sum _{i=1}^n(\overline{y}-\alpha -\beta \overline{x})(\hat{\beta }-\beta )(x_i-\overline{x})& =& (\overline{y}-\alpha -\beta \overline{x})(\hat{\beta }-\beta )\sum _{i=1}^n(x_i-\overline{x})\\ & =& (\overline{y}-\alpha -\beta \overline{x})(\hat{\beta }-\beta )(\sum _{i=1}^n{x}_i-\overline{x}\sum _{i=1}^{n}1)\\ & =& (\overline{y}-\alpha -\beta \overline{x})(\hat{\beta }-\beta )(n\overline{x}-n\overline{x})\cdots \sum _{i=1}^n{x}_i=n\overline{x}\\ & =& (\overline{y}-\alpha -\beta \overline{x})(\hat{\beta }-\beta )\cdot 0\\ & =& 0\end{array}$$