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, and also called the first moment. In general, E is not equal to f(E). Once the "average value" is known, one could then ask how far from this average value the values of X typically are, a question that is answered by the variance and standard deviation of a random variable. E can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of X.
Mathematically, this is known as the (generalised) problem of moments: for a given class of random variables X, find a collection {fi} of functions such that the expectation values E fully characterise the distribution of the random variable X.
Moments can only be defined for real-valued functions of random variables. 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 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 can be constructed; this uses the Iverson bracket, and has the value 1 if X has the value "green", 0 otherwise. Then, the expected value and other moments of this function can be determined.
Read more about this topic: Random Variable
Famous quotes containing the word moments:
“When it looks at great accomplishments, the world, bent on simplifying its images, likes best to look at the dramatic, picturesque moments experienced by its heroes.... But the no less creative years of preparation remain in the shadow.”
—Stefan Zweig (18811942)
“I like to compare the holiday season with the way a child listens to a favorite story. The pleasure is in the familiar way the story begins, the anticipation of familiar turns it takes, the familiar moments of suspense, and the familiar climax and ending.”
—Fred Rogers (20th century)
“Insults from an adolescent daughter are more painful, because they are seen as coming not from a child who lashes out impulsively, who has moments of intense anger and of negative feelings which are not integrated into that large body of responses, impressions and emotions we call ‘our feelings for someone,’ but instead they are coming from someone who is seen to know what she does.”
—Terri Apter (20th century)