(1)

\begin{align*} E(X+b) & =\sum_{i}(x_{i}+b)P(X=x_{i})\\ & =\sum_{i}x_{i}P(X=x_{i})+b\sum_{i}P(X=x_{i})\\ & =E(X)+b \end{align*}

(2)

\begin{align*} E(aX) & =\sum_{i}ax_{i}P(X=x_{i})\\ & =a\sum_{i}x_{i}P(X=x_{i})\\ & =aE(X) \end{align*}

(3)

\begin{align*} E(X+Y) & =\sum_{i,j}(x_{i}+y_{j})P(X=x_{i},Y=y_{j})\\ & =\sum_{i,j}x_{i}P(X=x_{i},Y=y_{j})+\sum_{i,j}y_{j}P(X=x_{i},Y=y_{j})\\ & =\sum_{i}x_{i}P(X=x_{i})+\sum_{j}y_{j}P(Y=y_{j})\\ & =E(X)+E(Y) \end{align*}

(4)

\begin{align*} Cov(X,Y) & =E\left(\left(X-E(X)\right)\left(Y-E(Y)\right)\right)\\ & =E(XY)-E\left(XE(Y)\right)-E\left(YE(X)\right)+E\left(E(X)E(Y)\right)\\ & =E(XY)-E(X)E(Y) \end{align*} これより、
\[ E(XY)=E(X)E(Y)+Cov(X,Y) \]