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A conditional expectation is the expectation of a random variable, given information on another variable or variables (mostly, by specifying their value).

A conditional expectation, or conditional mean, is the expectation of a random variable, given information on another variable or variables (mostly, by specifying their value). [The expected height of an adult US male is different from the expected height of an adult US male, born in 1942 to Micronesian parents.

For discrete variables, $\text{E}(X|Y=y) = \sum x \ \text{P}(X=x|Y=y)$, while for continuous variables, $\text{E}(X|Y=y)= \int x \ f_X(x|Y=y) \ dx$, where the sum and integral are over the possible values taken by $x$.

An expectation conditioned on some subset of the possible values taken by $Y$, such as $\text{E}(X\mid a < Y < b )$, say, is also a conditional expectation.

For an in-depth treatment of conditional expectations see:
Billingsley, P. (1995) "Probability and Measure", 3rd edition, John Wiley & Sons Inc., New York