離散型確率変数(discrete random variable) / 期待値(expected value)

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

X,Yが独立の場合

$$\begin{array}{rcl}E[XY]& =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}(x_i{y}_j)f_{XY}(x_i,y_j)}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}{x}_i{y}_j{f}_X(x_i)f_Y(y_i)\dots X,Y\mathrm{が}\mathrm{独}\mathrm{立}{f}_{XY}(x,y)=f_X(x)f_Y(y)}\\ & =& {\displaystyle \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}{x}_i{f}_X(x_i)y_j{f}_Y(y_i)}\\ & =& {\displaystyle (\sum _{i=1}^{\mathrm{\infty }}{x}_i{f}_X(x_i))(\sum _{j=1}^{\mathrm{\infty }}{y}_j{f}_Y(y_i))\dots \sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}{a}_i{b}_j=\sum _{i=1}^{\mathrm{\infty }}{a}_i\sum _{j=1}^{\mathrm{\infty }}{b}_j}\\ & =& E[X]E[Y]\end{array}$$