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.
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.
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.
Note that 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.
Note that 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
and conditional density of given the event
The conditional expectation of given is
When the denominator is zero, the expression is undefined.
Note that 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 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 .
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.
The Hilbert subspace
defined above is replaced with subsets thereof by restricting the functional form of g, rather than allowing any measureable 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, ¼, ½, ¾, 1; and
is the σ-algebra generated by the intervals with end-points 0, ½, 1. Here the conditional expectation is effectively the average over the minimal sets of the σ-algebra.
Consider the following:
- is a probability space.
- is a random variable on that probability space with finite expectation.
- is a sub-σ-algebra of .
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 .
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,
- A measurable space , and
- A random variable .
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
- 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 and for .
- 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 σ-algebra controls the "granularity" of the conditioning. A conditional expectation over a finer (larger) σ-algebra retains information about the probabilities of a larger class of events. A conditional expectation over a coarser (smaller) σ-algebra averages over more events.
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.
The Law of the unconscious statistician is then
This shows that conditional expectations are, like their unconditional counterparts, integrations,
against a conditional measure.