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
For two 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:
E[X|Y] = Σ E[X|Y=y] P(y)
E[X|Y=0] = (0) P(X=0|Y=0) + (1) P(X=1|Y=0) = P(X=1|Y=0)
E[X|Y=1] = (0) P(X=0|Y=1) + (1) P(X=1|Y=1) = P(X=1|Y=1)
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 conditional probabilities of X given Y.