連続型確率変数(continuous random variable) / 分散(variance)

$$\begin{array}{rcl}\mathrm{母}\mathrm{集}\mathrm{団}(population)\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{変}\mathrm{数}& :& X,Y\\ \mathrm{確}\mathrm{率}\mathrm{密}\mathrm{度}\mathrm{凾}\mathrm{数}(probabilitydensityfunction,PDF)& :& {\displaystyle {\int }_a^b{f}_X(x)\mathrm{d}x=P(a\le x\le b)}\\ & & {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_X(x)\mathrm{d}x=1}\\ \mathrm{累}\mathrm{積}\mathrm{分}\mathrm{布}\mathrm{凾}\mathrm{数}(cumulativedistributionfunction,CDF)& :& {\displaystyle F_X(x)={\int }_{-\mathrm{\infty }}^x{f}_X(t)\mathrm{d}t}\\ & & {\displaystyle f_X(x)=\frac{\mathrm{d}}{\mathrm{d}x}{F}_X(x)}\\ \mathrm{同}\mathrm{時}\mathrm{確}\mathrm{率}\mathrm{密}\mathrm{度}\mathrm{凾}\mathrm{数}(jointprobabilitydensityfunction)& :& {\displaystyle {\int }_a^b{\int }_c^d{f}_{XY}(x,y)\mathrm{d}x\mathrm{d}y=P(a\le x\le b,c\le y\le d)}\\ & & {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{XY}(x,y)\mathrm{d}x\mathrm{d}y=1}\\ \mathrm{周}\mathrm{辺}\mathrm{確}\mathrm{率}\mathrm{密}\mathrm{度}\mathrm{凾}\mathrm{数}(marginalprobabilitydensityfunction)& :& {\displaystyle f_X(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{XY}(x,y)\mathrm{d}y}\end{array}$$ $$\begin{array}{rcl}V[X]& \equiv & E[(X-E[X]{)}^2]\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x-E[X]{)}^2{f}_X(x)\mathrm{d}x}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x^2-2E[X]x+E[X{]}^2)f_X(x)\mathrm{d}x}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x^2{f}_X(x)-2E[X]x{f}_X(x)+E[X{]}^2{f}_X(x))\mathrm{d}x}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x^2)f_X(x)\mathrm{d}x-2E[X]{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x)f_X(x)\mathrm{d}x+E[X{]}^2{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_X(x)\mathrm{d}x}}}\\ & =& E[X^2]-2E[X]E[X]+E[X{]}^{2}1\\ & =& E[X^2]-2E[X{]}^2+E[X{]}^2\\ & =& E[X^2]-E[X{]}^2\\ V[cX]& =& E[(cX-E[cX]{)}^2]\dots c\mathrm{は}\mathrm{定}\mathrm{数}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(cx-E[cX]{)}^2{f}_X(x)\mathrm{d}x}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(cx-cE[X]{)}^2{f}_X(x)\mathrm{d}x}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(c(x-E[X]){)}^2{f}_X(x)\mathrm{d}x}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{c}^2(x-E[X]{)}^2{f}_X(x)\mathrm{d}x}\\ & =& c^2{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x-E[X]{)}^2{f}_X(x)\mathrm{d}x}\\ & =& c^{2}V[X]\\ V[X\pm t]& =& E[((X\pm t)-E[X\pm t]{)}^2]\dots t\mathrm{は}\mathrm{定}\mathrm{数}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}((x\pm t)-E[X\pm t]{)}^2{f}_X(x)\mathrm{d}x}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}((x\pm t)-(E[X]\pm t){)}^2{f}_X(x)\mathrm{d}x}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x\pm t-E[X]\mp t){)}^2{f}_X(x)\mathrm{d}x}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x-E[X]{)}^2{f}_X(x)\mathrm{d}x}\\ & =& V[X]\\ V[X\pm Y]& =& E[((X\pm Y)-E[X\pm Y]{)}^2]\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}((x\pm y)-E[X\pm Y]{)}^2{f}_{XY}(x,y)\mathrm{d}x\mathrm{d}y}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}((x\pm y)-(E[X]\pm E[Y]){)}^2{f}_{XY}(x,y)\mathrm{d}x\mathrm{d}y}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}((x-E[X])\pm (y-E[Y]){)}^2{f}_{XY}(x,y)\mathrm{d}x\mathrm{d}y}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}((x-E[X]{)}^2\pm 2(x-E[X])(y-E[Y])+(y-E[Y]{)}^2)f_{XY}(x,y)\mathrm{d}x\mathrm{d}y}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}((x-E[X]{)}^2{f}_{XY}(x,y)\pm 2(x-E[X])(y-E[Y])f_{XY}(x,y)+(y-E[Y]{)}^2{f}_{XY}(x,y))\mathrm{d}x\mathrm{d}y}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x-E[X]{)}^2{f}_{XY}(x,y)\mathrm{d}x\mathrm{d}y\pm 2{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x-E[X])(y-E[Y])f_{XY}(x,y)\mathrm{d}x\mathrm{d}y+{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(y-E[Y]{)}^2{f}_{XY}(x,y)\mathrm{d}x\mathrm{d}y}}}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x-E[X]{)}^2{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{XY}(x,y)\mathrm{d}y\mathrm{d}x\pm 2{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x-E[X])(y-E[Y])f_{XY}(x,y)\mathrm{d}x\mathrm{d}y+{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(y-E[Y]{)}^2{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{XY}(x,y)\mathrm{d}x\mathrm{d}y}}}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x-E[X]{)}^2{f}_X(x)\mathrm{d}x\pm 2{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x-E[X])(y-E[Y])f_{XY}(x,y)\mathrm{d}x\mathrm{d}y+{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(y-E[Y]{)}^2{f}_Y(y)\mathrm{d}y\dots \mathrm{周}\mathrm{辺}\mathrm{確}\mathrm{率}\mathrm{密}\mathrm{度}\mathrm{凾}\mathrm{数}\mathrm{を}\mathrm{適}\mathrm{用}}}}\\ & =& V[X]\pm 2Cov[X,Y]+V[Y]\dots Cov[X,Y]={\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x-E[X])(y-E[Y])f_{XY}(x,y)\mathrm{d}x\mathrm{d}y}\end{array}$$

X,Yが独立の場合

$$\begin{array}{rcl}V[X\pm Y]& =& {\displaystyle V[X]+V[Y]\dots X,Y\mathrm{が}\mathrm{独}\mathrm{立}\mathrm{な}\mathrm{の}\mathrm{で}Cov[X,Y]=0}\\ V[XY]& =& {\displaystyle E[(XY{)}^2]-E[XY{]}^2}\\ & =& {\displaystyle E[X^2{Y}^2]-E[XY{]}^2\dots (ab{)}^2=a^2{b}^2}\\ & =& {\displaystyle E[X^2]E[Y^2]-(E[X]E[Y]{)}^2\dots X,Y\mathrm{が}\mathrm{独}\mathrm{立}\mathrm{な}\mathrm{の}\mathrm{で}E[XY]=E[X]E[Y]}\\ & =& {\displaystyle E[X^2]E[Y^2]-E[X{]}^{2}E[Y{]}^2\dots (ab{)}^2=a^2{b}^2}\\ & =& {\displaystyle (V[X]+E[X{]}^2)(V[Y]+E[Y{]}^2)-E[X{]}^{2}E[Y{]}^2\dots V[X]=E[X^2]-E[X{]}^2\mathrm{よ}\mathrm{り}E[X^2]=V[X]+E[X{]}^2}\\ & =& {\displaystyle V[X]V[Y]+V[X]E[Y{]}^2+V[Y]E[X{]}^2+E[X{]}^{2}E[Y{]}^2-E[X{]}^{2}E[Y{]}^2}\\ & =& {\displaystyle V[X]V[Y]+V[X]E[Y{]}^2+V[Y]E[X{]}^2\dots X,Y\mathrm{が}\mathrm{独}\mathrm{立}\mathrm{で}\mathrm{も}V[X]V[Y]\mathrm{と}\mathrm{は}\mathrm{な}\mathrm{ら}\mathrm{な}\mathrm{い}}\end{array}$$