= Conditional Expectation = A '''conditional expectation''' is a [[Analysis/ExpectedValue|expectation]] given the realization of something. <> ---- == 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 === [[Analysis/BernoulliDistribution|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. {{attachment:expansion.svg}} But a more useful rewrite is ''E[X|Y] = p(X|Y)''. Then continue to evaluate the [[Statistics/ConditionalProbability|conditional probabilities]] of ''X'' given ''Y''. ---- CategoryRicottone