Conditional expectation
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.
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.
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 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.
[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.
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.
be continuous random variables with joint density
Not respecting this distinction can lead to contradictory conclusions as illustrated by the Borel-Kolmogorov paradox.
theory is, however, considered more intuitive[5] and admits important generalizations.
random variables, conditional expectation is also called regression.
minimizes the mean squared error: The conditional expectation of X is defined analogously, except instead of a single number
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 is 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
In the context of linear regression, this lack of uniqueness is called multicollinearity.
In the first example, the pushforward measure is a Dirac distribution at 1.
It can be shown that is a closed subspace of the Hilbert space
The conditional expectation is often approximated in applied mathematics and statistics due to the difficulties in analytically calculating it, and for interpolation.
These generalizations of conditional expectation come at the cost of many of its properties no longer holding.
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
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
, one can consider the collection of random variables It can be shown that they form a Markov kernel, that is, for almost all
[10][11] In this setting the conditional expectation is sometimes also denoted in operator notation as
, so we get that Thus the definition of conditional expectation is satisfied by the constant random variable
All random variables here are assumed without loss of generality to be non-negative.
By linearity the above property holds for simple functions: if
Combining the special case proved for simple functions, the definition of conditional expectation, and deploying the monotone convergence theorem: This holds for all
