|
Size: 1288
Comment: Initial commit
|
← Revision 6 as of 2026-07-28 02:45:08 ⇥
Size: 1333
Comment: Moving
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 1: | Line 1: |
| = Conditional Expectations = | = Conditional Expectation = |
| Line 3: | Line 3: |
| 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. |
| Line 11: | Line 11: |
| == Evaluation == | == Description == |
| Line 13: | Line 13: |
| For a discrete distribution, a conditional expectation is generally expanded as ''Σ E[X|Y=y] P(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. |
| Line 15: | Line 15: |
| For a continuous distribution, a conditional expectation is generally expanded as ''∫ E[X|Y=y] P(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''). |
| Line 17: | Line 17: |
| Then continue to evaluate the [[Statistics/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''). |
| Line 23: | Line 23: |
| 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: | [[Analysis/BernoulliDistribution|Bernoulli-distributed]] variables have some useful properties. |
| Line 25: | Line 25: |
| ''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]''. |
| Line 27: | Line 27: |
| ''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. |
| Line 29: | Line 29: |
| ''E[X|Y=1] = (0) P(X=0|Y=1) + (1) P(X=1|Y=1) = P(X=1|Y=1)'' | {{attachment:expansion.svg}} |
| Line 31: | Line 31: |
| ''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 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.