Conditional expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value given that a certain set of "conditions" is known to occur. In the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space.
With multiple random variables, if one random variable is mean independent of all others both individually and collectively, then each conditional expectation equals the random variable's (unconditional) expected value. This always holds if the variables are independent, but mean independence is a weaker condition.
Depending on the nature of the conditioning, the conditional expectation can be either a random variable itself or a fixed value. With two random variables, if the expectation of a random variable is expressed conditional on another random variable without a particular value of being specified, then the expectation of conditional on , denoted , is a function of the random variable and hence is itself a random variable. Alternatively, if the expectation of is expressed conditional on the occurrence of a particular value of , denoted , then the conditional expectation is a fixed value.
This concept generalizes to any probability space using measure theory.
In modern probability theory the concept of conditional probability is defined in terms of conditional expectation.
Contents
Examples
Example 1. 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. 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 Jan 1, 1990 to Dec 31, 1999. The unconditional expectation of rainfall for an unspecified day is the average of the rainfall amounts of 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.
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.^{[1]} In works of Paul Halmos^{[2]} and Joseph L. Doob^{[3]} from 1953, conditional expectation was generalized to its modern definition using subσalgebras.^{[4]}
Classical definition
Conditional expectation with respect to an event
In classical probability theory the conditional expectation of given an event (which may be the event for a random variable ) is the average of over all outcomes in , that is
Where is the cardinality of .
The sum above can be grouped by different values of , to get a sum over the range of
In modern^{[clarification needed]} probability theory, when is an event with strictly positive probability, it is possible to give a similar formula. This is notably the case for a discrete random variable and for in the range of if the event is . Let be a probability space, a random variable on that probability space, and an event with strictly positive probability . Then the conditional expectation of given the event is
where is the range of and is the probability measure defined, for each set , as , the conditional probability of given .
When (for instance if is a continuous random variable and is the event , this is in general the case), the Borel–Kolmogorov paradox demonstrates the ambiguity of attempting to define the conditional probability knowing the event . The above formula shows that this problem transposes to the conditional expectation. So instead one only defines the conditional expectation with respect to a σalgebra or a random variable.
Conditional expectation with respect to a random variable
If Y is a discrete random variable on the same probability space having range , then the conditional expectation of X with respect to Y is the random variable on defined by
There is a closely related function from to defined by
This function, which is different from the previous one, is the conditional expectation of X with respect to the σalgebra generated by Y. The two are related by
As mentioned above, if Y is a continuous random variable, it is not possible to define by this method. As explained in the Borel–Kolmogorov paradox, we have to specify what limiting procedure produces the set Y = y. If the event space has a distance function, then one procedure for doing so is as follows. Define the set . Assume that each is Pmeasurable and that for all . Then conditional expectation with respect to is welldefined. Take the limit as tends to 0 and define
Replacing this limiting process by the Radon–Nikodym derivative yields an analogous definition that works more generally.
Formal definition
Conditional expectation with respect to a subσalgebra
Consider the following:
 is a probability space.
 is a random variable on that probability space with finite expectation.
 is a subσalgebra of .
Since is a subalgebra 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 .^{[5]}
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
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,
 A measurable space , and
 A random variable .
Let be a measurable function such that, for every measurable function ,
Then the random variable , denoted as , is a conditional expectation of X given .
This definition is equivalent to defining the conditional expectation with respect to the subfield of (see above) defined by the preimage of Σ by Y. If we define
then
 .
Discussion
 This is not a constructive definition; we are merely given the required property that a conditional expectation must satisfy.
 The definition of may resemble that of for an event but these are very different objects. The former is a measurable function , while the latter is an element of . Evaluating the former at yields the latter.
 Existence of a conditional expectation function may be proven by the Radon–Nikodym theorem. A sufficient condition is that the (unconditional) expected value for X exists.
 Uniqueness can be shown to be almost sure: that is, versions of the same conditional expectation will only differ on a set of probability zero.
 The σalgebra controls the "granularity" of the conditioning. A conditional expectation over a finer (larger) σalgebra retains information about the probabilities of a larger class of events. A conditional expectation over a coarser (smaller) σalgebra averages over more events.
Conditioning as factorization
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In the definition of conditional expectation that we provided above, the fact that is a real random element is irrelevant. Let be a measurable space, where is a σalgebra on . A valued random element is a measurable function , i.e. for all . The distribution of is the probability measure defined as the pushforward measure , that is, such that .
Theorem. If is an integrable random variable, then there exists a unique integrable random element , defined almost surely, such that
for all .
Proof sketch. Let be such that . Then is a signed measure which is absolutely continuous with respect to . Indeed means exactly that , and since the integral of an integrable function on a set of probability 0 is 0, this proves absolute continuity. The Radon–Nikodym theorem then proves the existence of a density of with respect to . This density is .
Comparing with conditional expectation with respect to subσalgebras, it holds that
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:
The equation means that the integrals of and the composition over sets of the form , for , are identical.
This equation can be interpreted to say that the following diagram is commutative on average.
Computation
When X and Y are both discrete random variables, then the conditional expectation of X given the event Y = y can be considered as function of y for y in the range of Y:
where is the range of X.
If X is a continuous random variable, while Y remains a discrete variable, the conditional expectation is
with (where f_{X,Y}(x, y) gives the joint density of X and Y) being the conditional density of X given Y = y.
If both X and Y are continuous random variables, then the conditional expectation is
where (where f_{Y}(y) gives the density of Y).
Basic properties
All the following formulas are to be understood in an almost sure sense. The σalgebra could be replaced by a random variable .
 Pulling out independent factors:
 If is independent of , then .
Let . Then is independent of , so we get that
Thus the definition of conditional expectation is satisfied by the constant random variable , as desired.

 If is independent of , then . Note that this is not necessarily the case if is only independent of and of .
 If are independent, are independent, is independent of and is independent of , then .
 Stability:
 If is measurable, then .
 If Z is a random variable, then . In its simplest form, this says .
 Pulling out known factors:
 If is measurable, then .
 If Z is a random variable, then .
 Law of total expectation: .
 Tower property:
 For subσalgebras we have .
 A special case is when Z is a measurable random variable. Then and thus .
 Doob martingale property: the above with (which is measurable), and using also , gives .
 For random variables we have .
 For random variables we have .
 For subσalgebras we have .
 Linearity: we have and for .
 Positivity: If then .
 Monotonicity: If then .
 Monotone convergence: If then .
 Dominated convergence: If and with , then .
 Fatou's lemma: If then .
 Jensen's inequality: If is a convex function, then .

Conditional variance: Using the conditional expectation we can define, by analogy with the definition of the variance as the mean square deviation from the average, the conditional variance
 Definition:
 Algebraic formula for the variance:
 Law of total variance: .
 Martingale convergence: For a random variable , that has finite expectation, we have , if either is an increasing series of subσalgebras and or if is a decreasing series of subσalgebras and .
 Conditional expectation as projection: If are in the Hilbert space of squareintegrable real random variables (real random variables with finite second moment) then
 for measurable , we have , i.e. the conditional expectation is in the sense of the L^{2}(P) scalar product the orthogonal projection from to the linear subspace of measurable functions. (This allows to define and prove the existence of the conditional expectation based on the Hilbert projection theorem.)
 the mapping is selfadjoint:
 Conditioning is a contractive projection of L^{p} spaces . I.e., for any p ≥ 1.
 Doob's conditional independence property:^{[6]} If are conditionally independent given , then (equivalently, ).
See also
 Noncommutative conditional expectation
 Law of total probability
 Law of total expectation
 Law of total variance
 Law of total cumulance (generalizes the other three)
 Conditioning (probability)
 Joint probability distribution
 Disintegration theorem
Notes

^ Kolmogorov, Andrey (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung (in German). Berlin: Julius Springer. p. 46.^{[page needed]}
 Translation: Kolmogorov, Andrey (1956). Foundations of the Theory of Probability (2nd ed.). New York: Chelsea. ISBN 0828400237.^{[page needed]}
 ^ Oxtoby, J. C. (1953). "Review: Measure theory, by P. R. Halmos" (PDF). Bull. Amer. Math. Soc. 59 (1): 89–91. doi:10.1090/s000299041953096628.
 ^ J. L. Doob (1953). Stochastic Processes. John Wiley & Sons. ISBN 0471523690.
 ^ Olav Kallenberg: Foundations of Modern Probability. 2. edition. Springer, New York 2002, ISBN 0387953132, S. 573.
 ^ Billingsley, Patrick (1995). "Section 34. Conditional Expectation". Probability and Measure (3rd ed.). John Wiley & Sons. p. 445. ISBN 0471007102.
 ^ Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). York, PA, USA: Springer. p. 110. ISBN 0387953132.
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References
 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^{[page needed]}
 Grimmett, Geoffrey; Stirzaker, David (2001). Probability and Random Processes (3rd ed.). Oxford University Press. ISBN 0198572220., pages 67–69
External links
 Ushakov, N.G. (2001) [1994], "Conditional mathematical expectation", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104