Conditional Expectation for Discrete Random Variables

Given two discrete random variables $X$ and $Y,$ the conditional expectation of $Y$ given $X=x$ is the expected value of $Y$ computed using the conditional distribution of $Y$ given $X=x{:}$

$$E[Y \mid X = x] = \sum_y y \cdot p_{Y \mid X}(y \mid x).$$

This is the average value of $Y$ that we expect once we know $X$ has taken the specific value $x.$

For example, suppose $Y$ takes values in $\{1, 2, 3\}$ and its conditional distribution given $X = 5$ is

$$\begin{array}{c|ccc} y & 1 & 2 & 3 \\ \hline p_{Y \mid X}(y \mid 5) & 0.5 & 0.3 & 0.2 \end{array}$$

Then

$$\begin{align*} E[Y \mid X = 5] &= 1 \cdot 0.5 + 2 \cdot 0.3 + 3 \cdot 0.2 \\[3pt] &= 0.5 + 0.6 + 0.6 \\[3pt] &= 1.7. \end{align*}$$

Notice that the conditional probabilities in the row sum to $1$ — they form a valid pmf on the values of $Y.$