連続型確率変数(continuous random variable) / 期待値(expected value)

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

X,Yが独立の場合

$$\begin{array}{rcl}E[XY]& =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(xy)f_{XY}(x,y)\mathrm{d}x\mathrm{d}y}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}xy{f}_X(x)f_Y(y)\mathrm{d}x\mathrm{d}y\dots X,Y\mathrm{が}\mathrm{独}\mathrm{立}{f}_{XY}(x,y)=f_X(x)f_Y(y)}\\ & =& {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}x{f}_X(x)y{f}_Y(y)\mathrm{d}x\mathrm{d}y}\\ & =& {\displaystyle ({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}x{f}_X(x)\mathrm{d}x)({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}y{f}_Y(y)\mathrm{d}y)\dots {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)g(y)\mathrm{d}x\mathrm{d}y=({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)\mathrm{d}x)({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}g(y)\mathrm{d}y)}\\ & =& E[X]E[Y]\end{array}$$