In a probability distribution we need to consider the first, second, third, and fourth moments of the probability distribution.

This is going to be important for statistical physics and physical chemistry. And in these fields of science we are going to be able to calculate the number of states in a probability distribution which exist within a given distribution. 

Lets talk about moments. Such as second moment and second central moment.  Second central moment is a term used by science but a mathematician knows it only as variance.

From wikipedia:

See moment(mathematics).

Moment(physics) will talk about dipole moments and quadrapole moments, etc, which we all know and love. But these are not probability distributions.

https://en.m.wikipedia.org/wiki/Moment_(mat

In mathematics, the moments of a function are certain quantitative measures related to the shape of the function’s graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis.

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).