Editing Random variable (section)

==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 <math>\operatorname{E}[X]</math>, and also called the '''first [[Moment (mathematics)|moment]].''' In general, <math>\operatorname{E}[f(X)]</math> is not equal to <math>f(\operatorname{E}[X])</math>. Once the "average value" is known, one could then ask how far from this average value the values of <math>X</math> typically are, a question that is answered by the [[variance]] and [[standard deviation]] of a random variable. <math>\operatorname{E}[X]</math> can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of <math>X</math>.

Mathematically, this is known as the (generalised) [[problem of moments]]: for a given class of random variables <math>X</math>, find a collection <math>\{f_i\}</math> of functions such that the expectation values <math>\operatorname{E}[f_i(X)]</math> fully characterise the [[Probability distribution|distribution]] of the random variable <math>X</math>.

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 <math>f(X)=X</math> 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 variable|categorical]] random variable ''X'' that can take on the [[nominal data|nominal]] values "red", "blue" or "green", the real-valued function <math>[X = \text{green}]</math> can be constructed; this uses the [[Iverson bracket]], and has the value 1 if <math>X</math> has the value "green", 0 otherwise.  Then, the [[expected value]] and other moments of this function can be determined.