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| For two [[Statistics/BernoulliDistribution|Bernoulli-distributed]] variables (''X'' taking value 1 with probability ''p'' and value 0 with probability ''1-p''; ''Y'' taking value 1 with probability ''q'' and value 0 with probability ''1-q''), the expected value is evaluated as: | [[Statistics/BernoulliDistribution|Bernoulli-distributed]] variables have some useful properties. |
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| ''E[X|Y] = Σ E[X|Y=y] P(y)'' | 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]''. |
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| ''E[X|Y=0] = (0) P(X=0|Y=0) + (1) P(X=1|Y=0) = P(X=1|Y=0)'' | Given Bernoulli-distributed ''X'' and ''Y'', ''E[X|Y]'' can be evaluated by the above general expansion. |
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| ''E[X|Y=1] = (0) P(X=0|Y=1) + (1) P(X=1|Y=1) = P(X=1|Y=1)'' | {{attachment:expansion.svg}} |
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| ''E[X|Y] = P(X=1|Y=0) P(Y=0) + P(X=1|Y=1) P(Y=1) = P(X=1|Y=0) (1-q) + P(X=1|Y=1) (q)'' Then continue to evaluate the [[Statistics/ConditionalProbability|conditional probabilities]] of ''X'' given ''Y''. |
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''. |
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) (for all Y=y).
For a continuous distribution, a conditional expectation is generally expanded as ∫ E[X|Y=y] P(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.