information:highlighting, structural:scholar_edition, information:classification
Moments

The probability distribution of a random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. This is captured by the mathematical concept of expected value of a random variable, denoted E {\displaystyle \operatorname {E} }, and also called the first moment. In general, E {\displaystyle \operatorname {E} } is not equal to f (E) {\displaystyle f(\operatorname {E})}. Once the "average value" is known, one could then ask how far from this average value the values of X {\displaystyle X} typically are, a question that is answered by the variance and standard deviation of a random variable. E {\displaystyle \operatorname {E} } can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of X {\displaystyle X}.

Mathematically, this is known as the (generalised) problem of moments: for a given class of random variables X {\displaystyle X}, find a collection { f i } {\displaystyle \{f_{i}\}} of functions such that the expectation values E {\displaystyle \operatorname {E} } fully characterise the distribution of the random variable X {\displaystyle X}.

Moments can only be defined for real-valued functions of random variables (or complex-valued, etc.). If the random variable is itself real-valued, then moments of the variable itself can be taken, which are equivalent to moments of the identity function f (X) = X {\displaystyle f(X)=X} of the random variable. However, even for non-real-valued random variables, moments can be taken of real-valued functions of those variables. For example, for a categorical random variable X that can take on the nominal values "red", "blue" or "green", the real-valued function {\displaystyle } can be constructed; this uses the Iverson bracket, and has the value 1 if X {\displaystyle X} has the value "green", 0 otherwise. Then, the expected value and other moments of this function can be determined.