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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 analogously to conditional probability. The function form is either denoted or a separate function symbol such as is introduced with the meaning .


Example 1: Dice rolling

Consider the roll of a fair die 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.

1 2 3 4 5 6
A 0 1 0 1 0 1
B 0 1 1 0 1 0

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 falls in March. And 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.


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.[1] In works of Paul Halmos[2] and Joseph L. Doob[3] from 1953, conditional expectation was generalized to its modern definition using sub-σ-algebras.[4]


Conditioning on an event

If A is an event in 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 , 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 .

Conditioning on a discrete random variable is the same as conditioning on the corresponding event:

where A is the set .

Continuous random variables

Let and be continuous random variables with joint density 's density and conditional density of given the event The conditional expectation of given 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 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 , 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 theory is, however, considered more intuitive[5] and admits important generalizations. In the context of random variables, conditional expectation is also called regression.

In what follows let be a probability space, and in with mean and variance . The expectation minimizes the mean squared error:


The conditional expectation of X is defined analogously, except instead of a single number , the result will be a function . Let be a random vector. The conditional expectation is a measurable function such that


Note that unlike , the conditional expectation is not generally unique: there may be multiple minimizers of the mean squared error.


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 . 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.


The existence of a minimizer for is non-trivial. It can be shown that

is a closed subspace of the Hilbert space .[6] By the Hilbert projection theorem, the necessary and sufficient condition for to be a minimizer is that for all in M we have


In words, this equation says that the residual 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 .

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.[7]

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/ projection. If 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 distributions.

Conditional expectation with respect to a sub-σ-algebra

Conditional expectation with respect to a σ-algebra: in this example the probability space is the [0,1] interval with the Lebesgue measure. We define the following σ-algebras: ; is the σ-algebra generated by the intervals with end-points 0, 14, 12, 34, 1; and is the σ-algebra generated by the intervals with end-points 0, 12, 1. Here the conditional expectation is effectively the average over the minimal sets of the σ-algebra.

Consider the following:

Since is a sub -algebra of , the function is usually not -measurable, thus the existence of the integrals of the form , where and is the restriction of to , 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 , denoted as , is any -measurable function which satisfies:

for each .[8]

As noted in the discussion, this condition is equivalent to saying that the residual is orthogonal to the indicator functions :


The existence of can be established by noting that for is a finite measure on that is absolutely continuous with respect to . If is the natural injection from to , then is the restriction of to and is the restriction of to . Furthermore, is absolutely continuous with respect to , because the condition


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



Conditional probability

Main article: Regular 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 , is a probability measure.[9]

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 given is the up to a -nullset unique and integrable -valued -measurable random variable satisfying

for all .[10][11]

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 could be replaced by a random variable , i.e. .


Let . Then is independent of , so we get that

Thus the definition of conditional expectation is satisfied by the constant random variable , as desired.


For each we have , or equivalently

Since this is true for each , and both and are -measurable (the former property holds by definition; the latter property is key here), from this one can show

And this implies almost everywhere.


All random variables here are assumed without loss of generality to be non-negative. The general case can be treated with .

Fix and let . 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 is a simple function then .

Now let be -measurable. Then there exists a sequence of simple functions converging monotonically (here meaning ) and pointwise to . Consequently, for , the sequence converges monotonically and pointwise to .

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.

See also

Probability laws


  1. ^ Kolmogorov, Andrey (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung (in German). Berlin: Julius Springer. p. 46.
  2. ^ Oxtoby, J. C. (1953). "Review: Measure theory, by P. R. Halmos" (PDF). Bull. Amer. Math. Soc. 59 (1): 89–91. doi:10.1090/s0002-9904-1953-09662-8.
  3. ^ J. L. Doob (1953). Stochastic Processes. John Wiley & Sons. ISBN 0-471-52369-0.
  4. ^ Olav Kallenberg: Foundations of Modern Probability. 2. edition. Springer, New York 2002, ISBN 0-387-95313-2, p. 573.
  5. ^ "probability - Intuition behind Conditional Expectation". Mathematics Stack Exchange.
  6. ^ Brockwell, Peter J. (1991). Time series : theory and methods (2nd ed.). New York: Springer-Verlag. ISBN 978-1-4419-0320-4.
  7. ^ Hastie, Trevor. The elements of statistical learning : data mining, inference, and prediction (PDF) (Second, corrected 7th printing ed.). New York. ISBN 978-0-387-84858-7.
  8. ^ Billingsley, Patrick (1995). "Section 34. Conditional Expectation". Probability and Measure (3rd ed.). John Wiley & Sons. p. 445. ISBN 0-471-00710-2.
  9. ^ Klenke, Achim. Probability theory : a comprehensive course (Second ed.). London. ISBN 978-1-4471-5361-0.
  10. ^ Da Prato, Giuseppe; Zabczyk, Jerzy (2014). Stochastic Equations in Infinite Dimensions. Cambridge University Press. p. 26. doi:10.1017/CBO9781107295513. (Definition in separable Banach spaces)
  11. ^ Hytönen, Tuomas; van Neerven, Jan; Veraar, Mark; Weis, Lutz (2016). Analysis in Banach Spaces, Volume I: Martingales and Littlewood-Paley Theory. Springer Cham. doi:10.1007/978-3-319-48520-1. (Definition in general Banach spaces)
  12. ^ "Conditional expectation". Retrieved 2020-09-11.
  13. ^ Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). York, PA, USA: Springer. p. 110. ISBN 0-387-95313-2.


  • William Feller, An Introduction to Probability Theory and its Applications, vol 1, 1950, page 223
  • Paul A. Meyer, Probability and Potentials, Blaisdell Publishing Co., 1966, page 28
  • Grimmett, Geoffrey; Stirzaker, David (2001). Probability and Random Processes (3rd ed.). Oxford University Press. ISBN 0-19-857222-0., pages 67–69