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Conditional Expectation

The conditional expectation of an integrable random variable X given a sigma-algebra G is a G-measurable random variable, denoted E[X|G], such that

int_GE[X|G]dP=int_GXdP

for every G in G. It is unique up to sets of probability zero.

When conditioning on a random variable Y, the notation E[X|Y] means conditioning on the sigma-algebra generated by Y. Conditional expectation is the probability-theoretic analog of averaging while retaining specified information.

See also

Expectation Value, Random Variable

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References

Billingsley, P. Probability and Measure, 3rd ed. New York: Wiley, 1995.

Cite this as:

Weisstein, Eric W. "Conditional Expectation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConditionalExpectation.html

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