Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and simplified enough for intended applications.

This paper introduces novel formulas for binomial moments in the form of

The novel approach is a combinatorial argument coupled with clever algebraic simplifications which rely on symmetrization theory. As an interesting byproduct

The binomial distribution

Despite this large body of work on approximate inference, little is known about the

While the textbooks usually cover only the variance, sometimes also the skewness and kurtosis), there have been only few research papers discussing formulas for binomial moments of order

The discussed approaches still do not offer a satisfactory answer, as the formulas are not handy enough to be directly applicable. The author of the most general formula in [

Addressing the aforementioned issues with approaches in prior works, this paper offers the following novel contributions to computing binomial moments:

For code and examples visit the OSF repository

The remainder of the paper is organized as follows: the necessary background is given in Section

A random variable

Let

To state the results, we need

To work out the desired polynomial formulas, we need some standard algebraic notation. By

Below we discuss the contributions in more detail, deferring proofs to the end part of the paper. We denote

The first result is the derivation of the closed-form formula for raw binomial moments. While this formula appears in prior works [

The two proofs appear respectively in Section

Formulas for Raw Binomial Moments

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Table

Central Moments of Binomial Distribution. As above we denote

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The

To illustrate how useful is our

The estimate is uniformly sharp in all parameters; for the special case when

We have seen that the variance-based representation in Theorem

Algorithm

Variance Formula for Central Binomial Moments

In turn, Algorithm

Variance Formula for Central Binomial Moments

The proof is based on the fact that the

Then it remains to connect factorial moments to standard moments, or in other terms: factorial powers to powers. It is well known (see, for example, the discussion in [

Now Theorem

Here we take a direct approach, writing

From Equation (

By inspecting the products

As in the proof of Theorem

Throughout the proof, we will use the elementary estimates

By applying Remark

We now move on to the lower bound. The idea is to fix

Let us write

When

The above two bounds, in view of Theorem

This paper introduces novel and simpler formulas for binomial moments, derived by a combinatorial argument coupled with clever algebraic simplification which relies on symmetrization. An important application leads to sharp asymptotics for the growth of central binomial moments. Moreover, explicit algorithms and the working implementation are provided.

The implementation with examples is also available at

The author thanks to Peter Occil for correcting a mistaken inequality that occurred in the earlier version of this manuscript, and the anonymous reviewer for comments that leaded to a simplified version of the asymptotic bound.