Conditional expectations, continuous random variables

Summary

• \(X,Y\) are two continuous random variables.
• Given: partial information about the experiment, \(X=x\) ,
• The distribution of \(Y\) will then change from \(f_Y\left( y \right)\) to \(f_{Y|X}\left( x \right)\) and the expected value of \(Y\) will change from

\[E\left( Y \right)=\int_{-∞}^{∞}{ yf_Y\left( y \right)dy }\]

• to

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

• The latter is called the conditional expectation of \(Y\) given \(X=x\) .
• \(E\left( X=x \right)\) is a function of \(x\) which we may denote as \(g(x)\) .
• The conditional expectation of \(Y\) given \(X\) , \(E\left( X \right)\) , is then defined as the random variable \(g(X)\) .

Example

• \(f\left( x,y \right)=x+y\) for \(0≤x≤ 1\) and \(0≤y≤ 1\) . Suppose that we observe \(X=x\) . The distribution of \(Y\) will then change from

\[f_Y\left( y \right)=y+ \frac{1}{2}\]

• to

\[f_{Y|X}\left( x \right)= \frac{x+y}{x+ \frac{1}{2}}\]

• The expected value of \(Y\) will change from

\[\int_{0}^{1}{ y\left( y+ \frac{1}{2} \right)dy }= \frac{7}{12}\]

• to

\[\int_{0}^{1}{ y\left( \frac{x+y}{x+ \frac{1}{2}} \right)dy }= \frac{3x+2}{6x+3}=g\left( x \right)\]

• \(E\left( X \right)\) is then the random variable

\[Z= \frac{3X+2}{6X+3}\]

• with range \([5/9, 2/3]\) . It is possible to show that

\[f_Z\left( z \right)= \frac{1}{18{\left( 2z-1 \right)}^3}\]