Conditional expectation

In probability theory, it may be tempting to say that a conditional expectation is simple the expecation of a conditional probability distribution. Thus if X is a random variable, and A is an event whose probability is not 0, then the conditional probability distribution of X given A assigns a probability P(X ≤ x | A) to the interval from − ∞ to x, and we have a conditional probability distribution, which may have a first moment, called E(X | A), the conditional expected value of X given A.

However, that account omits some matters of interest and utility. If Y is another random variable, then the conditional expected value E(X | Y = y) of X given the event that Y = y is a function of y, which let us call g(y). Then the conditional expectation E(X | Y) is g(Y), another random variable whose value depends on that of 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.)

If X has an expected value, or -- what is the same thing -- E(|X|) < ∞, then the conditional expectation E(X | Y) also has an expected value, which is the same as that of X. That fact is the law of total expectation. See also law of total variance and law of total probability.

There is also the notion of conditional expected value of X given a sigma-algebra G, denoted E(X | G). This is a random variable that is G-measurable and whose integral over any G-measurable set is the same as the integral of X over the same set. If X happens to be G-measurable, then E(X | G) = X. The existence of this conditional expectation follows from the Radon-Nikodym theorem.