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

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

• to

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

• 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 $f_Z\left( 0.2 \right)=0.5$ , $f_Z\left( 0.8 \right)=0.5$ .