# Conditional Expectation Function

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A Conditional Expectation Function is an expectation function (for a random variable) computed with respect to the random variables conditional probability distribution.

**AKA:**Conditional Expectation.**Context:****range:**Conditional Expected Value.- It can be produced by a Conditional Expectation Function Creation Task.

**See:**Conditional Likelihood, Marginal Likelihood.

## References

### 2015

- http://www.statlect.com/xpvcnd1.htm
- QUOTE: The conditional expectation (or conditional mean, or conditional expected value) of a random variable is the expected value of the random variable itself, computed with respect to its conditional probability distribution. As in the case of the expected value, giving a completely rigorous definition of conditional expected value requires a complicated mathematical apparatus. To make things simpler, we do not give a completely rigorous definition in this lecture. We rather give an informal definition and we show how conditional expectation can be computed. In particular, we discuss how to compute the expected value of a random variable X when we observe the realization of another random variable Y, i.e. when we receive the information that Y=y. The expected value of X conditional on the information that Y=y is called conditional expectation of X given Y=y.
…

**Definition**Let [eq28] be the conditional distribution function of X given Y=y. The conditional expectation of X given Y=y is [eq29] where the integral is a Riemann-Stieltjes integral and the expected value exists and is well-defined only as long as the integral is well-defined.

- QUOTE: The conditional expectation (or conditional mean, or conditional expected value) of a random variable is the expected value of the random variable itself, computed with respect to its conditional probability distribution. As in the case of the expected value, giving a completely rigorous definition of conditional expected value requires a complicated mathematical apparatus. To make things simpler, we do not give a completely rigorous definition in this lecture. We rather give an informal definition and we show how conditional expectation can be computed. In particular, we discuss how to compute the expected value of a random variable X when we observe the realization of another random variable Y, i.e. when we receive the information that Y=y. The expected value of X conditional on the information that Y=y is called conditional expectation of X given Y=y.

### 1982

- (Bierens, 1982) ⇒ Herman J. Bierens. (1982). “Consistent Model Specification Tests.” In: Journal of Econometrics, 20(1).
- QUOTE: … The null hypothesis is that the regression function equals the conditional expectation function, which is tested against the alternative hypothesis that the null is false. These tests are based on a Fourier transform characterization of conditional expectations.