${\displaystyle E\left(aX+bY\right)=aE\left(X\right)+bE\left(Y\right)}$

${\displaystyle \mu =\left(100+500\right)\times \frac{1}{4}+\left(100+0\right)\times \frac{1}{4}+\left(0+500\right)\times \frac{1}{4}+\left(0+0\right)\times \frac{1}{4}}$

${\displaystyle =100\times \frac{1}{4}+500\times \frac{1}{4}+100\times \frac{1}{4}+0\times \frac{1}{4}+0\times \frac{1}{4}+500\times \frac{1}{4}+0\times \frac{1}{4}+0\times \frac{1}{4}}$

${\displaystyle =\left(100\times \frac{1}{4}+100\times \frac{1}{4}+0\times \frac{1}{4}+0\times \frac{1}{4}\right)+\left(500\times \frac{1}{4}+0\times \frac{1}{4}+500\times \frac{1}{4}+0\times \frac{1}{4}\right)}$

${\displaystyle =100\times \left(\frac{1}{4}+\frac{1}{4}\right)+0\times \left(\frac{1}{4}+\frac{1}{4}\right)+500\times \left(\frac{1}{4}+\frac{1}{4}\right)+0\times \left(\frac{1}{4}+\frac{1}{4}\right)}$

${\displaystyle =100\times \frac{1}{2}+0\times \frac{1}{2}+500\times \frac{1}{2}+0\times \frac{1}{2}}$

${\displaystyle =50+250}$

${\displaystyle =300}$

${\displaystyle E\left(100X+500Y\right)={\displaystyle \sum _{i=1}^2{\displaystyle \sum _{j=1}^2\left(100x_i+500y_j\right)h\left(x_i,y_j\right)}}}$

${\displaystyle ={\displaystyle \sum _{i=1}^2\left\{\left(100x_i+500y_1\right)h\left(x_i,y_1\right)+\left(100x_i+500y_2\right)h\left(x_i,y_2\right)\right\}}}$

${\displaystyle \begin{array}{l}=\left(100x_1+500y_1\right)h\left(x_1,y_1\right)+\left(100x_1+500y_2\right)h\left(x_1,y_2\right)\\ +\left(100x_2+500y_1\right)h\left(x_2,y_1\right)+\left(100x_2+500y_2\right)h\left(x_2,y_2\right)\end{array}}$

${\displaystyle \begin{array}{l}=100x_{1}h\left(x_1,y_1\right)+500y_{1}h\left(x_1,y_1\right)+100x_{1}h\left(x_1,y_2\right)+500y_{2}h\left(x_1,y_2\right)\\ +100x_{2}h\left(x_2,y_1\right)+500y_{1}h\left(x_2,y_1\right)+100x_{2}h\left(x_2,y_2\right)+500y_{2}h\left(x_2,y_2\right)\end{array}}$

${\displaystyle \begin{array}{l}=100x_{1}h\left(x_1,y_1\right)+100x_{1}h\left(x_1,y_2\right)+100x_{2}h\left(x_2,y_1\right)+100x_{2}h\left(x_2,y_2\right)\\ +500y_{1}h\left(x_1,y_1\right)+500y_{1}h\left(x_2,y_1\right)+500y_{2}h\left(x_1,y_2\right)+500y_{2}h\left(x_2,y_2\right)\end{array}}$

${\displaystyle ={\displaystyle \sum _{i=1}^2{\displaystyle \sum _{j=1}^{2}100x_{i}h\left(x_i,y_j\right)}}+{\displaystyle \sum _{i=1}^2{\displaystyle \sum _{j=1}^{2}500y_{j}h\left(x_i,y_j\right)}}}$

${\displaystyle ={\displaystyle \sum }_{i=1}^2\left\{100x_i{\displaystyle \sum }_{j=1}^{2}h\left(x_i,y_j\right)\right\}+{\displaystyle \sum }_{j=1}^2\left\{500y_j{\displaystyle \sum }_{i=1}^{2}h\left(x_i,y_j\right)\right\}}$

${\displaystyle {\displaystyle \sum }_{j=1}^{2}h\left(x_i,y_j\right)={\displaystyle \sum }_{j=1}^{2}P\left(X=x_i,Y=y_j\right)=P\left(X=x_i\right)=f\left(x_i\right)}$

${\displaystyle {\displaystyle \sum }_{i=1}^{2}h\left(x_i,y_j\right)={\displaystyle \sum }_{i=1}^{2}P\left(X=x_i,Y=y_j\right)=P\left(Y=y_i\right)=g\left(y_j\right)}$

${\displaystyle E\left(aX+bY\right)={\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^m\left(a{x}_i+b{y}_j\right)h\left(x_i,y_j\right)}}}$

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

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

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

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

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