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

$$\begin{array}{rcl} 母集団(population)の確率変数&:&X,Y\\ 確率密度凾数(probability \, density \, function, \, PDF)&:&\displaystyle\int_{a}^{b}f_X(x)\mathrm{d}x=P(a \leq x \leq b)\\ &&\displaystyle\int_{-\infty}^{\infty}f_X(x)\mathrm{d}x=1\\ 累積分布凾数(cumulative \, distribution \, function, \, CDF)&:&\displaystyle F_X(x)=\int_{-\infty}^{x}f_X(t)\mathrm{d}t\\ &&\displaystyle f_X(x)=\frac{\mathrm{d}}{\mathrm{d}x}F_X(x)\\ 同時確率密度凾数(joint\,probability\,density\,function)&:&\displaystyle\int_{a}^{b}\int_{c}^{d}f_{XY}(x, y)\mathrm{d}x\mathrm{d}y=P(a \leq x \leq b,\,c \leq y \leq d)\\ &&\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{XY}(x, y)\mathrm{d}x\mathrm{d}y=1\\ 周辺確率密度凾数(marginal\, probability\, density\, function)&:&\displaystyle f_X(x)=\int_{-\infty}^{\infty}f_{XY}(x,y)\mathrm{d}y \end{array}$$ $$\begin{array}{rcl} E[g(X)]&\equiv&\displaystyle\int_{-\infty}^{\infty} g(x) f_X(x) \mathrm{d}x\\ E[X]&=&\displaystyle\int_{-\infty}^{\infty} (x) f_X(x) \mathrm{d}x\\ E[cX] &=&\displaystyle\int_{-\infty}^{\infty} (c x) f_X(x) \mathrm{d}x\,\dotso\,cは定数\\ &=&c\displaystyle\int_{-\infty}^{\infty} x f_X(x) \mathrm{d}x\\ &=&c E[X] \\ E[X \pm t]&=&\displaystyle\int_{-\infty}^{\infty} (x \pm t) f_X(x) \mathrm{d}x\,\dotso\,tは定数\\ &=&\displaystyle\int_{-\infty}^{\infty} (x f_X(x) \pm t f_X(x)) \mathrm{d}x\\ &=&\displaystyle\int_{-\infty}^{\infty} x f_X(x) \pm \displaystyle\int_{-\infty}^{\infty} t f_X(x) \mathrm{d}x\\ &=&\displaystyle\int_{-\infty}^{\infty} x f_X(x) \pm t \displaystyle\int_{-\infty}^{\infty} f_X(x) \mathrm{d}x\\ &=&E[X] \pm t\\ E[X \pm Y]&=&\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} (x \pm y) f_{XY}(x, y) \mathrm{d}x\mathrm{d}y\\ &=&\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} (x f_{XY}(x, y) \pm y f_{XY}(x, y)) \mathrm{d}x\mathrm{d}y\\ &=&\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} x f_{XY}(x, y) \mathrm{d}x\mathrm{d}y \pm \displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} y f_{XY}(x, y)\mathrm{d}x\mathrm{d}y\\ &=&\displaystyle\int_{-\infty}^{\infty} x \int_{-\infty}^{\infty} f_{XY}(x, y) \mathrm{d}y\,\mathrm{d}x \pm \displaystyle\int_{-\infty}^{\infty} y \int_{-\infty}^{\infty} f_{XY}(x, y)\mathrm{d}x\,\mathrm{d}y\\ &=&\displaystyle\int_{-\infty}^{\infty} x f_{X}(x) \mathrm{d}x \pm \displaystyle\int_{-\infty}^{\infty} y f_{Y}(y) \mathrm{d}y\,\dotso\,周辺確率密度凾数を適用\\ &=& E[X] \pm E[Y] \\ \end{array}$$

X,Yが独立の場合

$$\begin{array}{rcl} E[XY]&=&\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(x y)f_{XY}(x, y)\mathrm{d}x\mathrm{d}y\\ &=&\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}xyf_{X}(x)f_{Y}(y)\mathrm{d}x\mathrm{d}y \,\dotso\,X,Yが独立\,f_{XY}(x, y)=f_{X}(x)f_{Y}(y)\\ &=&\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}xf_{X}(x)yf_{Y}(y)\mathrm{d}x\mathrm{d}y\\ &=&\displaystyle\left(\int_{-\infty}^{\infty}xf_{X}(x)\mathrm{d}x\right)\left(\int_{-\infty}^{\infty}yf_{Y}(y)\mathrm{d}y\right) \,\dotso\,\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x)g(y)\mathrm{d}x\mathrm{d}y=\left(\int_{-\infty}^{\infty}f(x)\mathrm{d}x\right)\left(\int_{-\infty}^{\infty}g(y)\mathrm{d}y\right)\\ &=&E[X]E[Y] \end{array}$$