${\displaystyle E\left(X+Y\right)=E\left(X\right)+E\left(Y\right)}$

${\displaystyle {\displaystyle \sum }_{j=1}^{n}h\left(x_i,y_j\right)}$${\displaystyle ={\displaystyle \sum }_{j=1}^{n}P\left(X=x_i,Y=y_j\right)}$${\displaystyle =P\left(X=x_i\right)=f\left(x_i\right)}$　･･････(3)

${\displaystyle {\displaystyle \sum }_{i=1}^{m}h\left(x_i,y_j\right)}$${\displaystyle ={\displaystyle \sum }_{i=1}^{m}P\left(X=x_i,Y=y_j\right)}$${\displaystyle =P\left(Y=y_i\right)=g\left(y_j\right)}$　･･････(4)

${\displaystyle E\left(X+Y\right)}$ ${\displaystyle ={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n\left(x_i+y_j\right)h\left(x_i,y_j\right)}}}$

${\displaystyle ={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n\left\{x_{i}h\left(x_i,y_j\right)+y_{j}h\left(x_i,y_j\right)\right\}}}}$

${\displaystyle ={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{x}_{i}h\left(x_i,y_j\right)}}}$ ${\displaystyle +{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{y}_{j}h\left(x_i,y_j\right)}}}$

${\displaystyle ={\displaystyle \sum _{i=1}^m\left\{x_i{\displaystyle \sum _{j=1}^{n}h\left(x_i,y_j\right)}\right\}}}$ ${\displaystyle +{\displaystyle \sum _{j=1}^n\left\{y_j{\displaystyle \sum _{i=1}^{m}h\left(x_i,y_j\right)}\right\}}}$

(3)，(4)より

${\displaystyle ={\displaystyle \sum _{i=1}^m{x}_{i}f\left(x_i\right)}+{\displaystyle \sum _{j=1}^n{y}_{j}g\left(y_j\right)}}$

${\displaystyle =E\left(X\right)+E\left(X\right)}$