離散型確率変数(discrete random variable) / 分散(variance)

$$\begin{array}{rcl}\mathrm{母}\mathrm{集}\mathrm{団}(population)\mathrm{の}\mathrm{確}\mathrm{率}\mathrm{変}\mathrm{数}& :& X,Y\\ \mathrm{確}\mathrm{率}\mathrm{質}\mathrm{量}\mathrm{凾}\mathrm{数}(probabilitymassfunction,PMF)& :& f_X(x)=P(X=x_i)\\ & & {\displaystyle \sum _{i=1}^{\mathrm{\infty }}{f}_X(x_i)=1}\\ \mathrm{累}\mathrm{積}\mathrm{分}\mathrm{布}\mathrm{凾}\mathrm{数}(cumulativedistributionfunction,CDF)& :& {\displaystyle F_X(x)=\sum _{i:x_i\le x}{f}_X(x_i)}\\ \mathrm{同}\mathrm{時}\mathrm{確}\mathrm{率}\mathrm{質}\mathrm{量}\mathrm{凾}\mathrm{数}(jointprobabilitymassfunction)& :& f_X(x,y)=P(X=x_i,Y=y_j)\\ & & {\displaystyle \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}{f}_{XY}(x_i,y_j)=1}\\ \mathrm{周}\mathrm{辺}\mathrm{確}\mathrm{率}\mathrm{質}\mathrm{量}\mathrm{凾}\mathrm{数}(marginalprobabilitymassfunction)& :& {\displaystyle f_X(x)=\sum _{j=1}^{\mathrm{\infty }}{f}_{XY}(x,y_i)}\end{array}$$ $$\begin{array}{rcl}V[X]& \equiv & E[(X-E[X]{)}^2]\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}(x_i-E[X]{)}^2{f}_X(x_i)}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}(x_i^2-2E[X]x_i+E[X{]}^2)f_X(x_i)}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}(x_i^2{f}_X(x_i)-2E[X]x_i{f}_X(x_i)+E[X{]}^2{f}_X(x_i))}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}(x_i^2)f_X(x_i)-2E[X]{\displaystyle \sum _{i=1}^{\mathrm{\infty }}(x_i)f_X(x_i)+E[X{]}^2{\displaystyle \sum _{i=1}^{\mathrm{\infty }}{f}_X(x_i)}}}\\ & =& 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 \sum _{i=1}^{\mathrm{\infty }}(c{x}_i-E[cX]{)}^2{f}_X(x_i)}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}(c{x}_i-cE[X]{)}^2{f}_X(x_i)}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}(c(x_i-E[X]){)}^2{f}_X(x_i)}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}{c}^2(x_i-E[X]{)}^2{f}_X(x_i)}\\ & =& c^2{\displaystyle \sum _{i=1}^{\mathrm{\infty }}(x_i-E[X]{)}^2{f}_X(x_i)}\\ & =& 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 \sum _{i=1}^{\mathrm{\infty }}((x_i\pm t)-E[X\pm t]{)}^2{f}_X(x_i)}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}((x_i\pm t)-(E[X]\pm t){)}^2{f}_X(x_i)}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}(x_i\pm t-E[X]\mp t){)}^2{f}_X(x_i)}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}(x_i-E[X]{)}^2{f}_X(x_i)}\\ & =& V[X]\\ V[X\pm Y]& =& E[((X\pm Y)-E[X\pm Y]{)}^2]\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}((x_i\pm y_j)-E[X\pm Y]{)}^2{f}_{XY}(x_i,y_j)}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}((x_i\pm y_j)-(E[X]\pm E[Y]){)}^2{f}_{XY}(x_i,y_j)}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}((x_i-E[X])\pm (y_j-E[Y]){)}^2{f}_{XY}(x_i,y_j)}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}((x_i-E[X]{)}^2\pm 2(x_i-E[X])(y_j-E[Y])+(y_j-E[Y]{)}^2)f_{XY}(x_i,y_j)}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}((x_i-E[X]{)}^2{f}_{XY}(x_i,y_j)\pm 2(x_i-E[X])(y_j-E[Y])f_{XY}(x_i,y_j)+(y_j-E[Y]{)}^2{f}_{XY}(x_i,y_j))}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}(x_i-E[X]{)}^2{f}_{XY}(x_i,y_j)\pm 2{\displaystyle \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}(x_i-E[X])(y_j-E[Y])f_{XY}(x_i,y_j)+{\displaystyle \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}(y_j-E[Y]{)}^2{f}_{XY}(x_i,y_j)}}}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}(x_i-E[X]{)}^2\sum _{j=1}^{\mathrm{\infty }}{f}_{XY}(x_i,y_j)\pm 2{\displaystyle \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}(x_i-E[X])(y_j-E[Y])f_{XY}(x_i,y_j)+{\displaystyle \sum _{j=1}^{\mathrm{\infty }}(y_j-E[Y]{)}^2\sum _{i=1}^{\mathrm{\infty }}{f}_{XY}(x_i,y_j)}}}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}(x_i-E[X]{)}^2{f}_X(x_i)\pm 2{\displaystyle \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}(x_i-E[X])(y_j-E[Y])f_{XY}(x_i,y_j)+{\displaystyle \sum _{j=1}^{\mathrm{\infty }}(y_j-E[Y]{)}^2{f}_Y(y_j)\dots \mathrm{周}\mathrm{辺}\mathrm{確}\mathrm{率}\mathrm{質}\mathrm{量}\mathrm{凾}\mathrm{数}\mathrm{を}\mathrm{適}\mathrm{用}}}}\\ & =& V[X]\pm 2Cov[X,Y]+V[Y]\dots Cov[X,Y]={\displaystyle \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}(x_i-E[X])(y_j-E[Y])f_{XY}(x_i,y_j)}\end{array}$$

X,Yが独立の場合

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