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The expected values of the powers of X are called the moments of X; the moments about the mean of X are expected values of powers of X − E[X]. The moments of some random variables can be used to specify their distributions, via their moment generating functions.
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 ...
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.
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 random graph on given vertices may be represented as a matrix of random variables, whose values specify the adjacency matrix of the random graph. A random function F {\displaystyle F} may be represented as a collection of random variables F ( x ) {\displaystyle F(x)} , giving the function's values at the various points x {\displaystyle x} in ...
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!).
Note: The conditional expected values E( X | Z) and E( Y | Z) are random variables whose values depend on the value of Z. Note that the conditional expected value of X given the event Z = z is a function of z. If we write E( X | Z = z) = g(z) then the random variable E( X | Z) is g(Z). Similar comments apply to the conditional covariance.
The expected value or mean of a random vector is a fixed vector [] whose elements are the expected values of the respective random variables. [ 3 ] : p.333 E [ X ] = ( E [ X 1 ] , . . .