Expected value of a function of several random variables

Summary

Discrete random variables

• $X,Y$ are two discrete random variables . The range of $X$ is ${x_1,…,x_n}$ and the range of $Y$ is ${y_1,…,y_m}$ . They have joint pmf $f\left( x,y \right)$ and $Z=g(X,Y)$ for some function $g$ . Then

$$E\left( Z \right)=\sum_{i=1}^{n}{ \sum_{j=1}^{m}{ g\left( x_i,y_j \right)f\left( x_i,y_j \right) } }$$

Example

• $Z=XY$ and $f(x,y)$ is defined by the following table:

$$E\left( Z \right)=\sum_{i=1}^{2}{ \sum_{j=1}^{2}{ x_iy_jf\left( x_i,y_j \right) } }=1⋅1⋅0.4+1⋅2⋅0.1+2⋅1⋅0.1+2⋅2⋅0.4=1.4$$

Continuous random variable

• $X,Y$ are two continuous random variables with joint pdf $f\left( x,y \right)$ and $Z=g(X,Y)$ for some function $g$ . Then

$$E\left( Z \right)=\int_{-∞}^{∞}{ \int_{-∞}^{∞}{ g\left( x,y \right)f\left( x,y \right)dxdy } }$$

Example

• $f\left( x,y \right)=4xy$ for $0≤x≤ 1$ and $0≤y≤ 1$ and $Z=XY$ . Then

$$E\left( Z \right)=\int_{0}^{1}{ \int_{0}^{1}{ xy⋅4xy dxdy } }= \frac{4}{9}$$