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{eqnarray*}E[aX+bY]&=&\displaystyle\sum_{i}\displaystyle\sum_{j}\ (ax_{i}+by_{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{eqnarray*}

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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{eqnarray*}E[aX+bY]&=&\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\ (ax+bY)\cdot f(x,y)\ dxfy\\&=&a\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\ xf(x,y)\ dxdy+b\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\ yf(x,y)\ dxdy\\&=&a\int_{-\infty}^{\infty}\ xf_{X}(x)\ dx+b\int_{-\infty}^{\infty}\ yf_{Y}(y)\ dy\ \ \ \left(\ \text{∵}\ \int_{-\infty}^{\infty}\ f(x, y)\ dy=f_{X}(x), \ \int_{-\infty}^{\infty}\ f(x, y)\ dx=f_{Y}(y)\right)\\&=&aE[X]+bE[Y]\end{eqnarray*}

Notice that \(f_{X}(x)\) and \(f_{Y}(y)\) are the marginal probability functions.