= Conditional Expectations =

A '''conditional expectation''' is the estimated outcome of an event given expectations of another event. The math notation is ''E[X|Y]''.

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== Evaluation ==

For a discrete distribution, a conditional expectation is generally expanded as ''Σ E[X|Y=y] p(Y=y)'' (for all ''Y=y'').

For a continuous distribution, a conditional expectation is generally expanded as ''∫ E[X|Y=y] p(Y=y) dx'' (for all ''Y=y'').

Then continue to evaluate the [[Statistics/ExpectedValues|expected values]] of ''X'' given ''Y=y''.



=== Bernoulli ===

[[Statistics/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''.



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