The concept of **moment** in mathematics evolved from the concept of **moment** in physics. The *n*th moment of a real-valued function *f*(*x*) of a real variable is

If (lower-case) *f* is a probability density function, then the value integral above is called the *n*th moment of the probability distribution. More generally, if (capital) *F* is a cumulative probability distribution function of any probability distribution, which may not have a density function, then the *n*th moment of the probability distribution is given by the Riemann-Stieltjes integral

The *n*th **central moment** of the probability distribution of a random variable *X* is

The central momemts are clearly translation-invariant, i.e., the *n*th central moment of *X* is the same as that of *X* + *c* for any constant *c* (in this context "constant" means a *non-random* quantity).

The first moment and the second and third *central* moments are linear in the sense that

The central moments beyond the third lack this linearity; in that respect they differ from the cumulants (the first three cumulants are the same as the first moment and the second and third *central* moments; the higher cumulants have a more complicated relationship with the central moments).