Conditional expectations, discrete random variables

Summary

\(X,Y\) are two discrete random variables with range \({x_1,…,x_n}\) and \({y_1,…,y_m}\) respectively.
Suppose that we have partial information about the experiment, we know the outcome of \(X\) , \(X=x\) .
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

\[\sum_{j=1}^{m}{ y_jf_Y\left( y_j \right) }\]

\[\sum_{j=1}^{m}{ y_jf_{Y|X}\left( x \right) }\]

The latter is called the conditional expectation of \(Y\) given \(X=x\) , \(E\left( X=x \right)\) .
\(E\left( X=x \right)\) 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

\(f(x,y)\) is defined by the following table:
\(f_Y\left( 0 \right)=0.5\) and \(f_Y\left( 1 \right)=0.5\) .
Suppose we observe \(X=0\) . Then \(f_{Y|X}\left( 0 \right)=0.4/0.5 =0.8\) and \(f_{Y|X}\left( 1 \right)=0.1/0.5=0.2\) . The expected value of \(Y\) changes from \(0.5\) to \(0.2\) . Thus, \(E\left( Y \right)=0.5\) and \(E\left( X=0 \right)=0.2.\) Similarly, \(E\left( X=1 \right)=0.8\) .
Define \(g\left( x \right)=E\left( X=x \right)\) and we have \(g\left( 0 \right)=0.2\) and \(g\left( 1 \right)=0.8\) .
Define \(Z\) as the random variable \(Z=E\left( X \right)=g\left( X \right)\) . \(Z\) takes the value \(0.2\) when \(X=0\) and \(0.8\) when \(X=1\) . The range of \(Z\) is \(\{ 0.2,0.8 \}\) and the pmf is given by \(f_Z\left( 0.2 \right)=0.5\) , \(f_Z\left( 0.8 \right)=0.5\) .