In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space. Depending on the context, the conditional expectation can be either a random variable or a function. The random variable is denoted E(X\mid Y) analogously to conditional probability. The function form is either denoted or a separate function symbol such as f(y) is introduced with the meaning.

Examples

Example 1: Dice rolling

Consider the roll of a fair and let A = 1 if the number is even (i.e., 2, 4, or 6) and A = 0 otherwise. Furthermore, let B = 1 if the number is prime (i.e., 2, 3, or 5) and B = 0 otherwise. The unconditional expectation of A is, but the expectation of A conditional on B = 1 (i.e., conditional on the die roll being 2, 3, or 5) is , and the expectation of A conditional on B = 0 (i.e., conditional on the die roll being 1, 4, or 6) is. Likewise, the expectation of B conditional on A = 1 is, and the expectation of B conditional on A = 0 is.

Example 2: Rainfall data

Suppose we have daily rainfall data (mm of rain each day) collected by a weather station on every day of the ten–year (3652-day) period from January 1, 1990, to December 31, 1999. The unconditional expectation of rainfall for an unspecified day is the average of the rainfall amounts for those 3652 days. The conditional expectation of rainfall for an otherwise unspecified day known to be (conditional on being) in the month of March, is the average of daily rainfall over all 310 days of the ten–year period that fall in March. Similarly, the conditional expectation of rainfall conditional on days dated March 2 is the average of the rainfall amounts that occurred on the ten days with that specific date.

History

The related concept of conditional probability dates back at least to Laplace, who calculated conditional distributions. It was Andrey Kolmogorov who, in 1933, formalized it using the Radon–Nikodym theorem. In works of Paul Halmos and Joseph L. Doob from 1953, conditional expectation was generalized to its modern definition using sub-σ-algebras.

Definitions

Conditioning on an event

If A is an event in \mathcal{F} with nonzero probability, and X is a discrete random variable, the conditional expectation of X given A is where the sum is taken over all possible outcomes of X. If P(A) = 0, the conditional expectation is undefined due to the division by zero.

Discrete random variables

If X and Y are discrete random variables, the conditional expectation of X given Y is where is the joint probability mass function of X and Y. The sum is taken over all possible outcomes of X. Remark that as above the expression is undefined if P(Y=y) = 0. Conditioning on a discrete random variable is the same as conditioning on the corresponding event: where A is the set { Y = y }.

Continuous random variables

Let X and Y be continuous random variables with joint density Y's density f_{Y}(y), and conditional density of X given the event Y=y. The conditional expectation of X given Y=y is When the denominator is zero, the expression is undefined. Conditioning on a continuous random variable is not the same as conditioning on the event { Y = y } as it was in the discrete case. For a discussion, see Conditioning on an event of probability zero. Not respecting this distinction can lead to contradictory conclusions as illustrated by the Borel-Kolmogorov paradox.

L2 random variables

All random variables in this section are assumed to be in L^2, that is square integrable. In its full generality, conditional expectation is developed without this assumption, see below under Conditional expectation with respect to a sub-σ-algebra. The L^2 theory is, however, considered more intuitive and admits important generalizations. In the context of L^2 random variables, conditional expectation is also called regression. In what follows let be a probability space, and in L^2 with mean \mu_X and variance \sigma_X^2. The expectation \mu_X minimizes the mean squared error: The conditional expectation of X is defined analogously, except instead of a single number \mu_X, the result will be a function e_X(y). Let be a random vector. The conditional expectation is a measurable function such that Note that unlike \mu_X, the conditional expectation e_X is not generally unique: there may be multiple minimizers of the mean squared error.

Uniqueness

Example 1: Consider the case where Y is the constant random variable that's always 1. Then the mean squared error is minimized by any function of the form Example 2: Consider the case where Y is the 2-dimensional random vector (X, 2X). Then clearly but in terms of functions it can be expressed as or or infinitely many other ways. In the context of linear regression, this lack of uniqueness is called multicollinearity. Conditional expectation is unique up to a set of measure zero in. The measure used is the pushforward measure induced by Y. In the first example, the pushforward measure is a Dirac distribution at 1. In the second it is concentrated on the "diagonal", so that any set not intersecting it has measure 0.

Existence

The existence of a minimizer for is non-trivial. It can be shown that is a closed subspace of the Hilbert space L^2(\Omega). By the Hilbert projection theorem, the necessary and sufficient condition for e_X to be a minimizer is that for all f(Y) in M we have In words, this equation says that the residual X - e_X(Y) is orthogonal to the space M of all functions of Y. This orthogonality condition, applied to the indicator functions , is used below to extend conditional expectation to the case that X and Y are not necessarily in L^2.

Connections to regression

The conditional expectation is often approximated in applied mathematics and statistics due to the difficulties in analytically calculating it, and for interpolation. The Hilbert subspace defined above is replaced with subsets thereof by restricting the functional form of g, rather than allowing any measurable function. Examples of this are decision tree regression when g is required to be a simple function, linear regression when g is required to be affine, etc. These generalizations of conditional expectation come at the cost of many of its properties no longer holding. For example, let M be the space of all linear functions of Y and let denote this generalized conditional expectation/L^2 projection. If M does not contain the constant functions, the tower property will not hold. An important special case is when X and Y are jointly normally distributed. In this case it can be shown that the conditional expectation is equivalent to linear regression: for coefficients described in Multivariate normal distribution.

Conditional expectation with respect to a sub-σ-algebra

Consider the following: Since \mathcal{H} is a sub \sigma-algebra of \mathcal{F}, the function is usually not \mathcal{H}-measurable, thus the existence of the integrals of the form, where and is the restriction of P to \mathcal{H}, cannot be stated in general. However, the local averages can be recovered in with the help of the conditional expectation. A conditional expectation of X given \mathcal{H}, denoted as, is any \mathcal{H}-measurable function which satisfies: for each. As noted in the L^2 discussion, this condition is equivalent to saying that the residual is orthogonal to the indicator functions 1_H:

Existence

The existence of can be established by noting that for is a finite measure on that is absolutely continuous with respect to P. If h is the natural injection from \mathcal{H} to \mathcal{F}, then is the restriction of \mu^X to \mathcal{H} and is the restriction of P to \mathcal{H}. Furthermore, is absolutely continuous with respect to P \circ h, because the condition implies Thus, we have where the derivatives are Radon–Nikodym derivatives of measures.

Conditional expectation with respect to a random variable

Consider, in addition to the above, The conditional expectation of X given Y is defined by applying the above construction on the σ-algebra generated by Y: By the Doob-Dynkin lemma, there exists a function such that

Discussion

Conditional probability

For a Borel subset B in, one can consider the collection of random variables It can be shown that they form a Markov kernel, that is, for almost all \omega, is a probability measure. The Law of the unconscious statistician is then This shows that conditional expectations are, like their unconditional counterparts, integrations, against a conditional measure.

General Definition

In full generality, consider: The conditional expectation of X given \mathcal{H} is the up to a P-nullset unique and integrable E-valued \mathcal{H}-measurable random variable satisfying for all. In this setting the conditional expectation is sometimes also denoted in operator notation as.

Basic properties

All the following formulas are to be understood in an almost sure sense. The σ-algebra \mathcal{H} could be replaced by a random variable Z, i.e.. Let. Then X is independent of 1_B, so we get that Thus the definition of conditional expectation is satisfied by the constant random variable E(X), as desired. \square For each we have, or equivalently Since this is true for each, and both and X are \mathcal{H}-measurable (the former property holds by definition; the latter property is key here), from this one can show And this implies almost everywhere. \square All random variables here are assumed without loss of generality to be non-negative. The general case can be treated with. Fix and let X = 1_A. Then for any Hence almost everywhere. Any simple function is a finite linear combination of indicator functions. By linearity the above property holds for simple functions: if X_n is a simple function then. Now let X be \mathcal{H}-measurable. Then there exists a sequence of simple functions converging monotonically (here meaning ) and pointwise to X. Consequently, for Y \geq 0, the sequence converges monotonically and pointwise to X Y. Also, since, the sequence converges monotonically and pointwise to Combining the special case proved for simple functions, the definition of conditional expectation, and deploying the monotone convergence theorem: This holds for all, whence almost everywhere. \square

Probability laws