Conditional expectation of a function of a random variable

Summary

Discrete random variables

• \(X,Y\) are two discrete random variables with range \({x_1,…,x_n}\) and \({y_1,…,y_m}\) respectively and \(h:R→R\) is an arbitrary function. We have

\[E\left( X=x \right)=\sum_{j=1}^{m}{ h\left( y_j \right)f_{Y|X}\left( x \right) }\]

Continuous random variables

• \(X,Y\) are two continuous random variables and \(h:R→R\) is an arbitrary function. We have

\[E\left( X=x \right)=\int_{-∞}^{∞}{ h\left( y \right)f_{Y|X}\left( y|x \right)dy }\]

Linear functions

• \(X,Y\) are two random variables and \(h:R→R\) is a linear function, \(h\left( y \right)=a+by\) . Then

\[E\left( X \right)=a+bE\left( X \right)\]

Conditional expectation of XY given Y

• \(E\left( X=x \right)\) is the number \(E\left( X=x \right)=xE\left( X=x \right)\)
• \(E\left( X \right)\) is the random variable \(XE\left( X \right)\)