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| = Conditional Expectations = | = Conditional Expectation = |
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| A '''conditional expectation''' is the estimated outcome of an event given expectations of another event. The math notation is ''E[X|Y]''. | A '''conditional expectation''' is a [[Analysis/ExpectedValue|expectation]] given the realization of something. |
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| == Evaluation == | == Description == |
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| For a discrete distribution, a conditional expectation is generally expanded as ''Σ E[X|Y=y] p(Y=y)'' (for all ''Y=y''). | 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. |
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| For a continuous distribution, a conditional expectation is generally expanded as ''∫ E[X|Y=y] p(Y=y) dx'' (for all ''Y=y''). | For a discrete probability distribution, a conditional expectation is generally evaluated as ''Σ E[X|Y=y] p(Y=y)'' (for all ''Y=y''). |
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| Then continue to evaluate the [[Analysis/ExpectedValues|expected values]] of ''X'' given ''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''). |
Conditional Expectation
A conditional expectation is a expectation given the realization of something.
Contents
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.