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The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. ... conditional expected value of X given Y ...
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 ...
The conditional expected value (), with a random variable, is not a simple number; it is a random variable whose value depends on the value of . That is, the conditional expected value of X {\displaystyle X} given the event Y = y {\displaystyle Y=y} is a number and it is a function of y {\displaystyle y} .
A number of special cases are given here. In the simplest case, where the random variable X takes on countably many values (so that its distribution is discrete), the proof is particularly simple, and holds without modification if X is a discrete random vector or even a discrete random element.
Note that the conditional expected value is a random variable in its own right, whose value depends on the value of . Notice that the conditional expected value of given the event = is a function of (this is where adherence to the conventional and rigidly case-sensitive notation of probability theory becomes important!).
Given any particular value y of the random variable Y, there is a conditional expectation (=) given the event Y = y. This quantity depends on the particular value y; it is a function () = (=).
If a random variable admits a density function, then the characteristic function is its Fourier dual, in the sense that each of them is a Fourier transform of the other. If a random variable has a moment-generating function (), then the domain of the characteristic function can be extended to the complex plane, and
The moment generating function of a real random variable is the expected value of , as a function of the real parameter . For a normal distribution with density f {\textstyle f} , mean μ {\textstyle \mu } and variance σ 2 {\textstyle \sigma ^{2}} , the moment generating function exists and is equal to