Significance of The Moments
The nth moment of a real-valued continuous function f(x) of a real variable about a value c is
It is possible to define moments for random variables in a more general fashion than moments for real values—see moments in metric spaces. The moment of a function, without further explanation, usually refers to the above expression with c = 0.
Usually, except in the special context of the problem of moments, the function f(x) will be a probability density function. The nth moment about zero of a probability density function f(x) is the expected value of Xn and is called a raw moment or crude moment. The moments about its mean μ are called central moments; these describe the shape of the function, independently of translation.
If f is a probability density function, then the value of the integral above is called the nth moment of the probability distribution. More generally, if F is a cumulative probability distribution function of any probability distribution, which may not have a density function, then the nth moment of the probability distribution is given by the Riemann–Stieltjes integral
where X is a random variable that has this cumulative distribution F, and E is the expectation operator or mean.
When
then the moment is said not to exist. If the nth moment about any point exists, so does (n − 1)th moment, and all lower-order moments, about every point.
The zeroth moment of any probability density function is 1, since the area under any probability density function must be equal to one.
| Moment number | Raw moment | Central moment | Standardized moment | Raw cumulant | Standardized cumulant |
|---|---|---|---|---|---|
| 1 | mean | 0 | 0 | mean | N/A |
| 2 | – | variance | 1 | variance | 1 |
| 3 | – | – | skewness | – | skewness |
| 4 | – | – | historical kurtosis | – | modern kurtosis (i.e. excess kurtosis} |
| 5+ | – | – | – | – | – |
Read more about this topic: Moment (mathematics)
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