For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from 0 to ∞) uniquely determines the distribution (Hausdorff moment problem).
The same is not true on unbounded intervals (Hamburger moment problem).
In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the moments of random variables.
The n-th moment of a real-valued continuous random variable with density function
The moment of a function, without further explanation, usually refers to the above expression with
are called central moments; these describe the shape of the function, independently of translation.
where X is a random variable that has this cumulative distribution F, and E is the expectation operator or mean.
These normalised central moments are dimensionless quantities, which represent the distribution independently of any linear change of scale.
The positive square root of the variance is the standard deviation
The normalised third central moment is called the skewness, often γ.
The fourth central moment is a measure of the heaviness of the tail of the distribution.
The kurtosis κ is defined to be the standardized fourth central moment.
For unbounded skew distributions not too far from normal, κ tends to be somewhere in the area of γ2 and 2γ2.
As with variance, skewness, and kurtosis, these are higher-order statistics, involving non-linear combinations of the data, and can be used for description or estimation of further shape parameters.
The higher the moment, the harder it is to estimate, in the sense that larger samples are required in order to obtain estimates of similar quality.
This is due to the excess degrees of freedom consumed by the higher orders.
Further, they can be subtle to interpret, often being most easily understood in terms of lower order moments – compare the higher-order derivatives of jerk and jounce in physics.
For example, just as the 4th-order moment (kurtosis) can be interpreted as "relative importance of tails as compared to shoulders in contribution to dispersion" (for a given amount of dispersion, higher kurtosis corresponds to thicker tails, while lower kurtosis corresponds to broader shoulders), the 5th-order moment can be interpreted as measuring "relative importance of tails as compared to center (mode and shoulders) in contribution to skewness" (for a given amount of skewness, higher 5th moment corresponds to higher skewness in the tail portions and little skewness of mode, while lower 5th moment corresponds to more skewness in shoulders).
is called the covariance and is one of the basic characteristics of dependency between random variables.
This identity follows by the convolution theorem for moment generating function and applying the chain rule for differentiating a product.
(These can also hold for variables that satisfy weaker conditions than independence.
This contrasts with the situation for central moments, whose computation uses up a degree of freedom by using the sample mean.
So for example an unbiased estimate of the population variance (the second central moment) is given by
in which the previous denominator n has been replaced by the degrees of freedom n − 1, and in which
A similar result even holds for moments of random vectors.
The n-th order lower and upper partial moments with respect to a reference point r may be expressed as
If the integral function does not converge, the partial moment does not exist.
The upside potential ratio may be expressed as a ratio of a first-order upper partial moment to a normalized second-order lower partial moment.
Let (M, d) be a metric space, and let B(M) be the Borel σ-algebra on M, the σ-algebra generated by the d-open subsets of M. (For technical reasons, it is also convenient to assume that M is a separable space with respect to the metric d.) Let 1 ≤ p ≤ ∞.
μ is said to have finite p-th central moment if the p-th central moment of μ about x0 is finite for some x0 ∈ M. This terminology for measures carries over to random variables in the usual way: if (Ω, Σ, P) is a probability space and X : Ω → M is a random variable, then the p-th central moment of X about x0 ∈ M is defined to be