Conditional Expectation

If $f_{X | Y} (x | y)$ is the conditional probability distribution/density function of $X$ given $Y = y$ at $x$ , the conditional expectation of $g (X)$ given $Y = y$ is

\begin{aligned} E [g (X) | Y = y] & = \sum_{x} g (x) f_{X | Y} (x | y) \\ or if continuous: \\ E [g (X) | Y = y] & = \int_{- \infty}^{\infty} g (x) f_{X | Y} (x | y) \end{aligned}

Letting $g (X) = X$ , the conditional mean of $X$ given $Y = y$ is

\begin{aligned} E (X | Y = y) & = \sum_{x} x f_{X | Y} (x | y) \\ E (X | Y = y) & = \int_{- \infty}^{\infty} x f_{X | Y} (x | y) d x \end{aligned}

and the second moment:

\begin{aligned} E (X^{2} | Y = y) & = \sum_{x} x^{2} f_{X | Y} (x | y) \\ E (X^{2} | Y = y) & = \int_{- \infty}^{\infty} x^{2} f_{X | Y} d x \end{aligned}

The conditional variance of $X$ given $Y = y$ is

V a r (X | Y = y) = E (X^{2} | Y = y) - [E (X | Y = y)]^{2}