How that the expectation of the sum of two random variables defined over the same sample space is the sum of the expectations. Hint: Let $p_1,p_2,....,p_n$be the probabilities associated with the n sample points; let$x_1,x_2,....,x_n$ and $y_1,y_2,....,y_n$, be the values of the random variables x and y for the n sample points. Write out $E\left(x\right),E\left(y\right),E(x+y)$.

Two random variables defined over the same sample space

The likelihood that a comparable continuous random variable has a value less than or equal to the function's argument is the value of the function.

Let $p_1,p_2,\mathrm{..},p_n$be the probabilities associated with the n sample points; let $x_1,x_2,\mathrm{..},x_n$and $y_1,y_2,\mathrm{..},y_n$, be the values of the random variables x and y for the n sample points.

The mean of the variable x and yare given below.

$\begin{array}{c}E\left(x\right)=\sum x_i{p}_i\\ E\left(x\right)=x_1{p}_1+x_2{p}_2+x_3{p}_3+\dots \\ E\left(y\right)=\sum x_i{p}_i\\ E\left(y\right)=y_1{p}_1+y_2{p}_2+y_3{p}_3+\dots \end{array}$

Let , the mean of variable is given below.

$\begin{array}{c}E\left(z_i\right)=\sum z_i{p}_i\\ E\left(z_i\right)=\sum \left(x_i+y_i\right)p_i\\ E\left(z_i\right)=\left(x_1+y_1\right)p_1+\left(x_2+y_2\right)p_2+\left(x_3+y_3\right)p_3+\cdots +\left(x_n+y_n\right)p_n\\ E\left(z_i\right)=x_1{p}_1+y_1{p}_1+x_2{p}_2+y_2{p}_2+\cdots +x_n{p}_n+y_n{p}_n\end{array}$

Solve further.

$\begin{array}{c}E\left(z_i\right)=x_1{p}_1+x_2{p}_2+x_3{p}_3+\cdots +x_n{p}_n+\cdots +y_1{p}_1+y_2{p}_2+y_3{p}_3+\cdots +x_n{p}_n\\ E\left(z_i\right)=\sum x_i{p}_i+\sum y_i{p}_i\\ E\left(z_i\right)=E\left(x\right)+E\left(y\right)\end{array}$

Hence, the value of expectation of variables are given below.

$\begin{array}{l}E\left(x\right)=x_1{p}_1+x_2{p}_2+x_3{p}_3+\dots \\ E\left(y\right)=y_1{p}_1+y_2{p}_2+y_3{p}_3+\dots \\ E\left(z_i\right)=E\left(x\right)+E\left(y\right)\end{array}$