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

$$\begin{array}{rcl} 母集団(population)の確率変数&:&X,Y\\ 確率密度凾数(probability \, density \, function, \, PDF)&:&\displaystyle\int_{a}^{b}f_X(x)\mathrm{d}x=P(a \leq x \leq b)\\ &&\displaystyle\int_{-\infty}^{\infty}f_X(x)\mathrm{d}x=1\\ 累積分布凾数(cumulative \, distribution \, function, \, CDF)&:&\displaystyle F_X(x)=\int_{-\infty}^{x}f_X(t)\mathrm{d}t\\ &&\displaystyle f_X(x)=\frac{\mathrm{d}}{\mathrm{d}x}F_X(x)\\ 同時確率密度凾数(joint\,probability\,density\,function)&:&\displaystyle\int_{a}^{b}\int_{c}^{d}f_{XY}(x, y)\mathrm{d}x\mathrm{d}y=P(a \leq x \leq b,\,c \leq y \leq d)\\ &&\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{XY}(x, y)\mathrm{d}x\mathrm{d}y=1\\ 周辺確率密度凾数(marginal\, probability\, density\, function)&:&\displaystyle f_X(x)=\int_{-\infty}^{\infty}f_{XY}(x,y)\mathrm{d}y \end{array}$$ $$\begin{array}{rcl} V[X]&\equiv&E[(X-E[X])^2]\\ &=&\displaystyle\int_{-\infty}^{\infty} (x-E[X])^2 f_X(x) \mathrm{d}x\\ &=&\displaystyle\int_{-\infty}^{\infty} (x^2-2E[X]x+E[X]^2) f_X(x) \mathrm{d}x\\ &=&\displaystyle\int_{-\infty}^{\infty} (x^2f_X(x)-2E[X]xf_X(x)+E[X]^2f_X(x)) \mathrm{d}x\\ &=&\displaystyle\int_{-\infty}^{\infty} (x^2) f_X(x)\mathrm{d}x -2E[X]\displaystyle\int_{-\infty}^{\infty} (x) f_X(x)\mathrm{d}x +E[X]^2\displaystyle\int_{-\infty}^{\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]\,\dotso\,cは定数\\ &=&\displaystyle\int_{-\infty}^{\infty} (c x - E[c X])^2 f_X(x) \mathrm{d}x\\ &=&\displaystyle\int_{-\infty}^{\infty} (c x - c E[X])^2 f_X(x) \mathrm{d}x\\ &=&\displaystyle\int_{-\infty}^{\infty} (c (x - E[X]))^2 f_X(x) \mathrm{d}x\\ &=&\displaystyle\int_{-\infty}^{\infty} c^2 (x-E[X])^2 f_X(x) \mathrm{d}x\\ &=&c^2\displaystyle\int_{-\infty}^{\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]\,\dotso\,tは定数\\ &=&\displaystyle\int_{-\infty}^{\infty} ((x \pm t)-E[X \pm t])^2 f_X(x) \mathrm{d}x\\ &=&\displaystyle\int_{-\infty}^{\infty} ((x \pm t)-(E[X] \pm t))^2 f_X(x) \mathrm{d}x\\ &=&\displaystyle\int_{-\infty}^{\infty} (x \pm t - E[X] \mp t))^2 f_X(x) \mathrm{d}x\\ &=&\displaystyle\int_{-\infty}^{\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_{-\infty}^{\infty}\int_{-\infty}^{\infty} ((x \pm y)-E[X \pm Y])^2 f_{XY}(x, y) \mathrm{d}x\mathrm{d}y\\ &=&\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} ((x \pm y)-(E[X] \pm E[Y]))^2 f_{XY}(x, y) \mathrm{d}x\mathrm{d}y\\ &=&\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} ((x - E[X]) \pm (y - E[Y]))^2 f_{XY}(x, y) \mathrm{d}x\mathrm{d}y\\ &=&\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\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_{-\infty}^{\infty}\int_{-\infty}^{\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_{-\infty}^{\infty}\int_{-\infty}^{\infty} (x - E[X])^2 f_{XY}(x, y) \mathrm{d}x\mathrm{d}y \pm 2\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(x - E[X])(y - E[Y]) f_{XY}(x, y) \mathrm{d}x\mathrm{d}y + \displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} (y - E[Y])^2 f_{XY}(x, y) \mathrm{d}x\mathrm{d}y\\ &=&\displaystyle\int_{-\infty}^{\infty} (x - E[X])^2 \int_{-\infty}^{\infty} f_{XY}(x, y) \mathrm{d}y\, \mathrm{d}x \pm 2\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(x - E[X])(y - E[Y]) f_{XY}(x, y) \mathrm{d}x\mathrm{d}y + \displaystyle\int_{-\infty}^{\infty} (y - E[Y])^2 \int_{-\infty}^{\infty} f_{XY}(x, y) \mathrm{d}x\,\mathrm{d}y\\ &=&\displaystyle\int_{-\infty}^{\infty} (x - E[X])^2 f_{X}(x) \mathrm{d}x \pm 2\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(x - E[X])(y - E[Y]) f_{XY}(x, y) \mathrm{d}x\mathrm{d}y + \displaystyle\int_{-\infty}^{\infty} (y - E[Y])^2 f_{Y}(y) \mathrm{d}y \,\dotso\,周辺確率密度凾数を適用\\ &=&V[X] \pm 2Cov[X,Y] +V[Y]\,\dotso\,Cov[X,Y]=\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\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

$$\begin{array}{rcl} V[X \pm Y]&=&\displaystyle V[X]+V[Y]\,\dotso\,X,Yが独立なのでCov[X,Y]=0\\ V[XY]&=&\displaystyle E[(XY)^2]-E[XY]^2\\ &=&\displaystyle E[X^2Y^2]-E[XY]^2\,\dotso\,(ab)^2=a^2b^2\\ &=&\displaystyle E[X^2]E[Y^2]-(E[X]E[Y])^2\,\dotso\,X,Yが独立なのでE[XY]=E[X]E[Y]\\ &=&\displaystyle E[X^2]E[Y^2]-E[X]^2E[Y]^2\,\dotso\,(ab)^2=a^2b^2\\ &=&\displaystyle (V[X]+E[X]^2)(V[Y]+E[Y]^2)-E[X]^2E[Y]^2\,\dotso\,V[X]=E[X^2]-E[X]^2よりE[X^2]=V[X]+E[X]^2\\ &=&\displaystyle V[X]V[Y]+V[X]E[Y]^2+V[Y]E[X]^2+E[X]^2E[Y]^2-E[X]^2E[Y]^2\\ &=&\displaystyle V[X]V[Y]+V[X]E[Y]^2+V[Y]E[X]^2\,\dotso\,X,Yが独立でもV[X]V[Y]とはならない\\ \end{array}$$