conditional expectation,
n. (Statistics) 1. the expected value of a random variable, X, or any function of it, given that an event, B, is known to have occurred; written E(X | B). This is the sum or integral of the products of all the possible values of the random variable or function and the respective conditional probabilities of each. Where yi is a value of the discrete random variable Y, E(X), the expected value of X, is the sum of the products of the conditional expectations E(X | Y=yi) and the probabilities of each of the yi respectively. See also Bayes's theorem, marginal expectation, Radon-Nikodym theorem. 2. (of a random variable f, given x) formally, the function
e(x)(f) = dσ/dα
on a product probability space (X × Y, μ), defined as the Radon-Nikodym derivative of σ with respect to α, where
σ(A) = ∫A×Y f dμ
and
α(A) = μ(A ×Y). 3. (of X, given X1,..., Xn) rigorously, any random variable g that is measurable with respect to the sigma-field D generated by all inverse images of Borel sets Bk, {Xk ∈ Bk}, and that satisfies
∫D g dP = ∫D X dP
for all D in D. This is written E(X | X1,..., Xn) or E(X | D), and the definition holds for any sigma-subfield D. One can then define
P(A | X) = E(χA | X).
If X and Y have a joint density f(x, y), then X has density f(x) and
E(Y | X = x) = ∫ y (f(x, y)/f(x)) dy.
e(x)(f) = dσ/dα
on a product probability space (X × Y, μ), defined as the Radon-Nikodym derivative of σ with respect to α, where
σ(A) = ∫A×Y f dμ
and
α(A) = μ(A ×Y). 3. (of X, given X1,..., Xn) rigorously, any random variable g that is measurable with respect to the sigma-field D generated by all inverse images of Borel sets Bk, {Xk ∈ Bk}, and that satisfies
∫D g dP = ∫D X dP
for all D in D. This is written E(X | X1,..., Xn) or E(X | D), and the definition holds for any sigma-subfield D. One can then define
P(A | X) = E(χA | X).
If X and Y have a joint density f(x, y), then X has density f(x) and
E(Y | X = x) = ∫ y (f(x, y)/f(x)) dy.