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The distribution of the de Moivre experiment

This paper introduces the $s$ -Bernoulli and bi $^s$ nomial distributions derived from de Moivre's random experiment, analyzes their probabilistic properties, and establishes their convergence to Poisson and normal distributions as $n \to \infty$ .

Original authors: Hacène Belbachir, Hamza Zeggada

Published 2026-06-16

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Hacène Belbachir, Hamza Zeggada

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are running a carnival game. In the classic version, you have a coin. You flip it, and it lands on either Heads (1) or Tails (0). If you flip it many times, the results follow a famous pattern called the Binomial Distribution. This is the "coin toss" model everyone knows.

Now, imagine upgrading that game. Instead of a two-sided coin, you have a special die with $s+1$ sides. Let's say the sides are numbered 0, 1, 2, up to $s$ . When you roll this die, you don't just get "success" or "failure"; you get a specific number of points.

This paper is about what happens when you roll this special die $n$ times and add up all the points you get. The authors, Belbachir and Zeggada, are exploring the mathematical rules that govern this "multi-sided" game, which they call the de Moivre experiment (named after an 18th-century mathematician who first looked at similar problems).

Here is a breakdown of their findings in simple terms:

1. The New "Dice" Rules (The s-Bernoulli Distribution)

First, they looked at a single roll of this special die.

The Old Way: A coin has two outcomes.
The New Way: This die has $s+1$ outcomes (0 through $s$ ).
The Twist: The chances of landing on each number aren't necessarily equal. The authors set up a system where the probability of getting a high number is linked to a "success" probability ( $p$ ) and a "failure" probability ( $q$ ).
The Result: They created a new formula (the s-Bernoulli distribution) to calculate the average score and how much the scores usually vary (variance) for this single roll. They also figured out how to calculate the "skewness" (is the curve lopsided?) and "kurtosis" (is the peak sharp or flat?).

2. The Big Score (The Bisnomial Distribution)

Next, they asked: "What happens if I roll this die $n$ times and add up the total points?"

In the coin world, adding up $n$ flips gives you the Binomial Distribution.
In this new world, adding up $n$ rolls of the $s+1$ sided die gives them the Bisnomial Distribution.
They proved that this new distribution has its own set of rules for calculating averages, variances, and complex patterns (moments). They even connected these patterns to some fancy math tools called Eulerian numbers and Bell polynomials, which are like secret codes that help mathematicians predict the shape of the data.

3. The Magic Transformation (Convergence)

The most exciting part of the paper is what happens when you play the game a huge number of times (as $n$ goes to infinity). The authors show that this complex "multi-sided die" game starts to look like two very famous, simpler patterns:

The Rare Event Transformation (Poisson):
If the game is set up so that getting a high score is very rare (like winning a lottery), but you play the game millions of times, the total number of high scores you get will eventually follow the Poisson Distribution.
- Analogy: Imagine counting how many times a specific, rare number comes up on your die over a million rolls. Even though the die is complex, the count of those rare hits settles into a simple, predictable pattern known as the Poisson distribution. The authors prove this happens even with their new "multi-sided" rules.
The Crowd Transformation (Normal Distribution):
If you play the game many times and the scores aren't too rare, the total sum of points starts to form a perfect Bell Curve (the Normal Distribution).
- Analogy: Think of a crowd of people. If you ask one person for their height, it's random. But if you ask a million people and average their heights, the results always form a smooth, symmetrical hill. The authors prove that no matter how many sides your die has, if you roll it enough times, the total score will always form this perfect hill. This is a version of the famous "Central Limit Theorem."

4. Why This Matters (According to the Paper)

The authors point out that while the "coin toss" (2 sides) has been studied for centuries, the "multi-sided die" (more than 2 sides) has been largely ignored in probability theory until now.

They are essentially saying: "We finally have the complete instruction manual for this specific type of dice game."
They provide the formulas to calculate the average, the spread, and the shape of the results.
They prove that even though this game is more complex than a coin flip, it still obeys the same ultimate laws of nature (converging to Poisson and Normal distributions) when played on a large scale.

In summary: The paper takes a complex, multi-sided dice game that no one had fully modeled before, writes down the rules for how to calculate the odds, and proves that if you play it long enough, it magically simplifies into the two most famous patterns in statistics: the Poisson distribution (for rare events) and the Normal distribution (the Bell Curve).

Technical Summary: The Distribution of the de Moivre Experiment

Problem Statement and Motivation
This paper addresses a historical gap in probability modeling regarding the "de Moivre experiment." While Abraham de Moivre, in his 1711 work The Doctrine of Chances, analyzed the combinatorial problem of determining the number of ways to obtain a specific sum $k$ when rolling $n$ dice with $(s+1)$ faces, the probabilistic distribution governing this experiment has not been formally developed in the literature. Historically, the case where the number of faces equals 2 (the coin toss) led to the Bernoulli and binomial distributions. However, the generalization to a uniform distribution on $\{0, 1, \dots, s\}$ (where $s > 1$ ) remained unmodeled until the simplified expressions for the associated coefficients were established in the 2000s. The authors aim to formalize this experiment, define the associated probability laws, and investigate their limiting behaviors.

Methodology and Definitions
The study proceeds by defining the fundamental random variable and its associated distributions:

The $s$ -Bernoulli Distribution: The authors define a random variable $X$ representing the outcome of a single die with $(s+1)$ faces, taking values in $\{0, 1, \dots, s\}$ . The probability mass function is parameterized by $p$ (probability of success) and $q$ (probability of failure), with specific constraints ensuring the sum of probabilities equals 1.
- The generating function $G_X(t)$ is derived for both cases where $p \neq q$ and $p = q$ .
- Explicit formulas for expectation and variance are provided.
- The paper establishes a relationship between the moments of this distribution and extended Eulerian numbers $A(n, k, s)$ . For the case $p=q$ , the moments are expressed using Bernoulli numbers.
The Bisnomial Distribution: By repeating the $s$ -Bernoulli experiment $n$ times independently, the authors define the bisnomial distribution, denoted $B^{(s)}(n, p)$ . The probability mass function involves the bisnomial coefficients $\binom{n}{k}_s$ , which count the number of ways to distribute $k$ points among $n$ dice with $(s+1)$ faces.
- The generating function for the bisnomial distribution is derived as the $n$ -th power of the $s$ -Bernoulli generating function.
- Formulas for the expectation and variance are established.
- Higher-order moments are expressed using partial Bell polynomials, linking the distribution to combinatorial structures.

Key Results: Limiting Distributions
The core theoretical contribution of the paper is the proof of convergence for the bisnomial distribution under specific limiting conditions as $n \to \infty$ .

Convergence to the Poisson Distribution:
The authors prove that if the number of trials $n$ increases indefinitely while the expected value of the sum remains constant ( $\lambda_s$ ), the bisnomial distribution converges to a Poisson distribution with parameter $\lambda_s$ .
- The proof utilizes the Implicit Function Theorem to approximate the relationship between the parameters $p_n$ and $q_n$ as $n \to \infty$ .
- By expanding these parameters in power series and applying Lemma 5.2 (which establishes the limit of the normalized bisnomial coefficient $\binom{n}{k}_s / n^k$ ), the authors demonstrate that the probability mass function converges to $\frac{\lambda_s^k}{k!} e^{-\lambda_s}$ .
- The paper notes that this result supports historical arguments that the Poisson distribution should be attributed to de Moivre, as he derived it as a limit of the binomial distribution in 1711, predating Poisson's 1837 work.
Convergence to the Normal Distribution:
The paper establishes that for large $n$ , the bisnomial distribution approximates a Normal distribution.
- Using the Moment Generating Function (MGF) method, the authors analyze the standardized variable $Z = \frac{X - E(X)}{\sigma_s}$ .
- Through Taylor series expansion of the MGF, they show that as $n \to \infty$ , the MGF of $Z$ converges to $e^{t^2/2}$ , the MGF of the standard normal distribution.
- This confirms that the bisnomial distribution satisfies a Central Limit Theorem analogous to the classical binomial case, with mean $\mu_s$ and variance $\sigma_s^2$ derived from the parameters of the $s$ -Bernoulli distribution.

Significance and Claims
The paper claims to provide the first comprehensive probabilistic modelization of the de Moivre experiment for a general number of faces $(s+1)$ . By introducing the $s$ -Bernoulli and bisnomial distributions, the authors extend the classical binomial framework to a broader combinatorial setting.

The significance of the work lies in:

Historical Context: It bridges the gap between de Moivre's 1711 combinatorial lemma and modern probability theory, formally defining the distribution laws that were previously implicit.
Mathematical Generalization: It generalizes the binomial distribution (where $s=1$ ) to the bisnomial case, providing explicit formulas for moments, generating functions, and skewness/kurtosis coefficients involving Eulerian and Bell polynomials.
Limiting Behavior: It rigorously proves that these generalized distributions converge to the Poisson and Normal distributions, thereby unifying the de Moivre experiment with the fundamental limit theorems of probability theory.

The authors maintain a modest scope, focusing strictly on the definition, properties, and limiting cases of these distributions without proposing new experimental applications or future implications beyond the mathematical proofs presented.

1. The New "Dice" Rules (The s-Bernoulli Distribution)

2. The Big Score (The Bisnomial Distribution)

3. The Magic Transformation (Convergence)

4. Why This Matters (According to the Paper)

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