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:
“Parenting forces us to get to know ourselves better than we ever might have imagined we could—and in many new ways. . . . We’ll discover talents we never dreamed we had and fervently wish for others at moments we feel we desperately need them. As time goes on, we’ll probably discover that we have more to give and can give more than we ever imagined. But we’ll also find that there are limits to our giving, and that may be hard for us to accept.”
—Fred Rogers (20th century)
“Self-expression is not enough; experiment is not enough; the recording of special moments or cases is not enough. All of the arts have broken faith or lost connection with their origin and function. They have ceased to be concerned with the legitimate and permanent material of art.”
—Jane Heap (c. 1880–1964)
“Within the circuit of this plodding life
There enter moments of an azure hue,
Untarnished fair as is the violet
Or anemone, when the spring strews them
By some meandering rivulet, which make
The best philosophy untrue that aims
But to console man for his grievances.
I have remembered when the winter came,”
—Henry David Thoreau (1817–1862)