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

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