What is a factorial moment?

The r-th factorial moment of a discrete random variable X is E[X^(r)] = E[X(X-1)(X-2)···(X-r+1)]. The 1st factorial moment equals the mean. The 2nd is E[X(X-1)], which combined with the mean gives the variance.

How is a factorial moment different from an ordinary moment?

An ordinary moment uses X^r, while a factorial moment uses the falling factorial X^(r) = X(X-1)···(X-r+1). Factorial moments simplify calculations involving combinations and are directly linked to the probability generating function via its derivatives at 1.

Do my probabilities need to sum exactly to 1?

Yes. The tool checks that your probabilities sum to 1 within a small tolerance (±0.001). If they do not, you will see an error. Rescale your values before calculating.

Can I use non-integer outcome values?

You can enter non-integer x values, but factorial moments are most meaningful for non-negative integer outcomes (count data). For non-integer x the falling factorial X(X-1)···(X-r+1) is still computed algebraically, but the statistical interpretation may not apply.

What is the falling factorial?

The falling factorial x^(r) (read "x to the r falling") equals x*(x-1)*(x-2)*...*(x-r+1), a product of r consecutive descending terms starting at x. For r=0 it equals 1 by convention. It counts the number of ways to arrange r items chosen from x distinct items.

Factorial Moment Calculator

Enter a discrete probability distribution and choose r to compute the r-th factorial moment instantly.

Distribution Input

Enter each outcome and its probability. Probabilities must sum to 1.

Outcome x Probability P(X=x)

Order r

Quick presets

Enter your distribution and click Calculate

Factorial Moments

Order r	Formula	Value

Derived Statistics

Distribution Summary

x	P(X=x)

Summary

The r-th factorial moment of a discrete random variable X is defined as E[X(X-1)(X-2)···(X-r+1)] = sum over all x of x*(x-1)*...*(x-r+1)*P(X=x). Factorial moments arise naturally in combinatorics, queuing theory, and the analysis of count data. They are especially convenient for deriving ordinary moments and for working with probability generating functions.

How it works

Enter each outcome value (x) and its probability P(X=x) as rows in the table.
Probabilities must be non-negative and sum to 1 (the tool validates this).
Select the order r (1 through 6) of the factorial moment you want.
Click "Calculate" to compute E[X(X-1)···(X-r+1)] = sum x_falling_r * P(X=x).
Results for all orders 1 through r are shown together for comparison.

Use cases

Compute the mean (1st factorial moment) of a count distribution.
Derive the variance using E[X(X-1)] and E[X]: Var(X) = E[X(X-1)] + E[X] - (E[X])^2.
Analyze Poisson, binomial, or negative-binomial distributions analytically.
Check probability generating function derivatives against empirical data.
Solve combinatorics problems involving ordered selections from a population.
Validate simulation output against theoretical factorial moments.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu