In probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution. Probability theory is the mathematical study of probability. ... In probability (and especially gambling), the expected value (or (mathematical) expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds... This article defines some terms which characterize probability distributions of two or more variables. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...

Special cases

In the simplest case, if A is an event whose probability is not 0, then

is a probability measure on A and E(X | A) is the expectation of X with respect to this probability P_A. In case X is a discrete random variable (that is a random variable which with probability 1 takes on only a countable number of values), and with finite first moment, the expectation is explicitly given by the infinite In mathematics, a probability space is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ... In mathematics the term countable set is used to describe the size of a set, e. ...

where {X = r} is the event that X takes on the value r. Since X has finite first moment, it can be shown this sum converges absolutely. Note that the sum is countable since {X = r} has probability 0 for only countable many values of r.

Note that if X is the indicator function of an event S then E(X | A) is just the conditional probability P_A(S).

If Y is another real random variable, then for each value of y we consider the event {Y = y}. (Reminder for those less-than-accustomed to the conventional language and notation of probability theory: this paragraph is an example of why case-sensitivity of notation must not be neglected, since capital Y and lower-case y refer to different things.) The conditional expectation E(X | Y = y) is shorthand for E(X | {Y = y}). Of course in general this may not be defined since {Y = y} may have zero probability.

The way out of this limitation is as follows: Note that if both X and Y are discrete random variables then for any subset B of Y The word discrete comes from the Latin word discretus which means separate. ...

where 1 is the indicator function. For general random variables Y, P{Y=r} is zero. As a first step in dealing with this problem, let us consider the case Y has a continuous distribution function. This means there is a non-negative integrable function φ_Y on R which is the density of Y. This means In the mathematical subfield of set theory, the indicator function is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ...

for any a in R. We can then show the following: for any integrable random variable X, there is a function g on R such that

This function g is a suitable candidate for the conditional expectation.

In order to handle the general case, we need more powerful mathematical machinery.

Mathematical formalism

Let X, Y be real random variables on some probability space (Ω, M, P) where M is the σ-algebra of measurable sets on which P is defined. We consider two measures on R:

• Q defined by Q(B) = P(Y⁻¹(B)) for every Borel subset B of R is a probability measure on the real line R. Now
• P_X given by

If X is an integrable random variable, then P_X is absolutely continuous with respect to Q. In this case, it can be shown the Radon-Nikodym derivative of P_X with respect to Q exists; moreover it is uniquely determined almost everywhere with respect to Q. This random variable is the conditional expectation of X given Y, or more accurately a version of the conditional expectation of X given Y. In mathematics, the Radon-Nikodym theorem is a result in functional analysis that states that if a measure Q is absolutely continuous with respect to another sigma-finite measure P then there is a measurable function f, taking values in [0,∞], on the underlying space such that for any measurable...

It follows that the conditional expectation satisfies

for any Borel subset B of R.

Conditioning as factorization

In the definition of conditional expectation that we provided above, the fact Y is a real random variable is irrelevant: Let U be a measurable space, that is a set equipped with a σ-algebra of subsets. A U-valued random variable is a function Y: Ω → U such that Y⁻¹(B) is an element of M for any measurable subset B of U.

We consider the measure Q on U given as above: Q(B) = P(Y⁻¹(B)) for every measurable subset B of U. Q is a probability measure on the measurable space U defined on its σ-algebra of measurable sets.

Theorem. If X is an integrable real random variable on Ω then there is one and up to equivalence a.e. relative to Q, only one integrable function g such that for any measurable subset B of U:

There are a number of ways of proving this; one as suggested above, is to note that the expression on the left hand side defines as a function of the set B a countably additive probability measure on the measurable subsets of U. Moreover, this measure is absolutely continuous relative to Q. Indeed Q(B) = 0 means exactly that Y⁻¹(B) has probability 0. The integral of an integrable function on a set of probability 0 is itself 0. This proves absolute continuity.

The defining condition of conditional expectation then is the equation

We can further interpret this equality by considering the abstract change of variables formula to transport the integral on the right hand side to an integral over Ω: In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ...

This equation can be interpreted to say that the following diagram is commutative in the average. In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

Image File history File links Download high resolution version (1177x799, 9 KB) A commutative diagram for en:conditional expectation. ...

The equation means that the integrals of X and the composition E(X|Y)ˆY over sets of the form Y⁻¹(B) for B measurable are identical.

Conditioning relative to a subalgebra

There is another viewpoint for conditioning involving σ-subalgebras N of the σ-algebra M. This version is a trivial specialization of the preceding: we simply take U to be the space Ω with the σ-algebra N and Y the identity map. We state the result:

Theorem. If X is an integrable real random variable on Ω then there is one and up to equivalence a.e. relative to P, only one integrable function g such that for any set B belonging to the subalgebra N

This form of conditional expectation is usually written: E(X|N). This version is preferred by probabilists. One reason is that on the space of square-integrable real random variables (in other words, real random variables with finite second moment) the mapping X → E(X|N) is the self-adjoint orthogonal projection In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ... In geometry, an orthogonal projection of a k-dimensional object onto a d-dimensional hyperplane (d < k) is obtained by intersections of (k − d)- dimensional hyperplanes drawn through the points of an object orthogonally to the d-hyperplane. ...

Basic properties

Let (Ω,M,P) be a probability space.

• Conditioning with respect to a σ-subalgebra N is linear on the space of integrable real random variables.
• E(1|N) = 1
• Jensen's inequality holds: If f is a convex function,then

• Conditioning is a contractive projection

for any s ≥1. A convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have Convex function on interval I.e. ...

See also

Law of total probability, Law of total expectation, Law of total variance, Law of total cumulance (This fourth item generalizes the other three.) Nomenclature in probability theory is not wholly standard. ... The proposition in probability theory known as the law of total expectation, or the law of iterated expectations, or perhaps by any of a variety of other names, states that if X is an integrable random variable (i. ... In probability theory, the law of total variance states that if X and Y are random variables on the same probability space, and the variance of X is finite, then In language perhaps better known to statisticians than to probabilists, the first term is the unexplained component of the variance... Main article: cumulant In probability theory and mathematical statistics, the law of total cumulance is a generalization to cumulants of the law of total probability, the law of total expectation, and the law of total variance. ...

References

• William Feller, An Introduction to Probability Theory and its Applications, vol 1, 1950
• Paul A. Meyer, Probability and Potentials, Blaisdell Publishing Co., 1966