Proof

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Discrete Case

Let f(x, y) be the joint probability function such that $f(x_i,y_j)=P(X=x_i,Y=y_j)$.

$$\begin{array}{rcl}E[aX+bY]& =& {\displaystyle \sum _i{\displaystyle \sum _j(a{x}_i+b{y}_j)\cdot f(x_i,y_j)}}\\ & =& a{\displaystyle \sum _i{\displaystyle \sum _j{x}_{i}f(x_i,y_j)+b{\displaystyle \sum _i{\displaystyle \sum _j{y}_{j}f(x_i,y_j)}}}}\\ & =& a{\displaystyle \sum _i{x}_i{f}_X(x_i)+b{\displaystyle \sum _j{y}_j{f}_Y(y_j)\left(\text{∵}{\displaystyle \sum _{j}f(x_i,y_j)=f_X(x_i),{\displaystyle \sum _{i}f(x_i,y_j)=f_Y(y_j)}}\right)}}\\ & =& aE[X]+bE[Y]\end{array}$$

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Continuous Case

Let f(x, y) be the probability function such that $f(x,y)=P(X=x,Y=y)$.

$$\begin{array}{rcl}E[aX+bY]& =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(ax+bY)\cdot f(x,y)dxfy\\ & =& a{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}xf(x,y)dxdy+b{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}yf(x,y)dxdy\\ & =& a{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}x{f}_X(x)dx+b{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}y{f}_Y(y)dy(\text{∵}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x,y)dy=f_X(x),{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x,y)dx=f_Y(y))\\ & =& aE[X]+bE[Y]\end{array}$$

Notice that $f_X(x)$ and $f_Y(y)$ are the marginal probability functions.