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| For a discrete distribution, a conditional expectation is generally expanded as ''Σ E[X|Y=y] P(y)'' (for all ''Y=y''). | For a discrete distribution, a conditional expectation is generally expanded as ''Σ E[X|Y=y] p(Y=y)'' (for all ''Y=y''). |
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| For a continuous distribution, a conditional expectation is generally expanded as ''∫ E[X|Y=y] P(y) dx'' (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''). |
Conditional Expectations
A conditional expectation is the estimated outcome of an event given expectations of another event. The math notation is E[X|Y].
Contents
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 expected values of X given 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.