Conditional Expectation
A conditional expectation is a expectation given the realization of something.
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Description
A conditional expectation is generally notated as E[X|Y], where the random variable (here X) sits on the left of the bar, and the realized values sit on the right. Sometimes the bar is replaced with a semicolon.
For a discrete probability distribution, a conditional expectation is generally evaluated as Σ E[X|Y=y] p(Y=y) (for all Y=y).
For a continuous probability distribution, a conditional expectation is generally evaluated as ∫ E[X|Y=y] p(Y=y) dx (for all Y=y).
Bernoulli
Bernoulli-distributed variables have some useful properties.
Given a Bernoulli-distributed X, for the same reason that E[X] = p(X=1) (i.e. the 0 term eliminates itself), it is also true that E[X|Y] = E[X=1|Y].
Given Bernoulli-distributed X and Y, E[X|Y] can be evaluated by the above general expansion.
But a more useful rewrite is E[X|Y] = p(X|Y). Then continue to evaluate the conditional probabilities of X given Y.