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Introduction of p-nomial Distribution as a Generalization of Binomial Distribution

Aziz ATTA

Mathematics and Structural Analysis, Atta Engineering Design Office, El Jadida, Morocco

Abstract: The theory of probability and statistics, thanks to its continuous modernization, has become more and more important in our life given its presence in several fields such as economics and prevision [8]. The binomial distribution is among the oldest probability distributions introduced by Bernoulli [1]. In the same context, we thought of generalizing this probability distribution under the name p-nomial distribution using p-nomial coefficients p-nomial theorem [7]. In this article, we are going to be interested in the introduction of this new probability distribution as well as an establishment of its various standard characteristics. the purpose of this article is therefore summarized in the tracing of the theoretical framework with some examples of application of the said p-nomial distribution.

Keywords: p-nomial coefficients, p-nomial identity, p-nomial distribution, probability tree, trinomial distribution

Received August 16, 2020; Revised September 17, 2020; Accepted September 24, 2020

Citation: Aziz ATTA, Introduction of p-nomial Distribution as a Generalization of Binomial Distribution. Turkish Journal of Analysis and Number Theory, 8 (5). 80-90. January 2020. doi:10.12691/tjant-8-5-1

1. Introduction

The binomial distribution is one of the oldest probability laws studied [1]. It was introduced by Jacques Bernoulli who referred to it in 1713 in his work Ars Conjectandi. Between 1708 and 1718, the multinomial distribution (multidimensional generalization of the binomial distribution), the negative binomial distribution as well as the approximation of the binomial distribution by the Poisson’s distribution, the law of large numbers for the binomial distribution and an approximation of the tail of the binomial distribution are discovered [2].

Thanks to the expression of its mass function, the binomial distribution has been used by several scientists to perform calculations in concrete situations. This is the case of Abraham de Moivre who succeeds in finding an approximation of the binomial distribution by the normal distribution. In 1812, Pierre-Simon de Laplace resumed this work. Francis Galton creates the Galton plate which allows a physical representation of this convergence [3]. In 1909, Émile Borel states and proves, in the case of binomial law, the first version of the strong law of large numbers [4].

Binomial law appeared in many applications in the 20th century [5]: in genetics, animal biology, plant ecology, for statistical tests, in different physical models such as telephone networks or the Ehrenfest’s urn model, etc.

The name “binomial” of this law comes from [6] the writing of its mass function (see below) which contains a binomial coefficient resulting from the development of the binomial . Indeed, if X is a random variable following a binomial law of characteristics n and p, then: ,

In the same sense, I thought of generalizing the binomial coefficients by calling them p-nomial coefficients; this is based on a generalization of Pascal's formula. After this definition, I introduced the notion of the p-nomial law as a generalization of the binomial distribution in probability.

In this article, we’ll recall the p-nomial coefficients that we have already defined [7]. We’ll also present the properties and characteristics of this distribution as well as some examples of application. Finally, we’ll be interested in the particular case of trinomial distribution.

Note that this article aims to introduce this new distribution and show its importance through a few examples. This introduction will contain as expected an establishment of the characteristics of this distribution such as its expectation, its variance and its moments.

2. p-nomial Coefficients

In this part, we will quickly recall the definition of p-nomial coefficients, their expression as well as the p-nomial theorem. For that, the reader interested in more details on these concepts can consult the article cited in the reference [7].

2.1. Definition

Let be a non-zero natural integer. We define p-nomial coefficient and we note the kth p-nomial coefficient among by the following recurrent relation:

Such that .

2.2. Expressions

Using the fundamental theorem [7], we can establish an expression for the p-nomial coefficients:

To simplify the writing, we introduce the symbol to group the summations. For :

So, the p-nomial coefficient is a combination of multinomial coefficients [13].

2.2. p-nomial Theorem

Let a non-zero natural integer. We have following equality [7]:

3. p-nomial Distribution

3.1. Definition

Let and two non-zero natural integers. A random variable follow a distribution where if:

Indeed, this distribution is well defined:

Which shows that: .

Moreover, using p-nomial theorem:

3.2. Probability Tree of the (p+1)-nomial Distribution

We consider p identical coins noted . We will launch the p coins at the same time for independent trials. Each outcome has a fixed probability, the same from trial to trial. Also, the probability to have a head for each coin is and to have a tail is .

For each trial, we count the number of heads and tails. To simplify, if we get heads , we will write the result as follows where H means heads and T means tails.

We note the kth trial. The possible results for each trial are . We represent below the probability tree:

Figure 1. Probability tree of the (p + 1)-nomial distribution

We count, after n trials, the total number of heads obtained. If we denote by X this random variable then the possible values of X are . In addition, if we note the probability of having a head , then the random variable X follows a distribution noted such that :

We note:

3.3. Probability Measure

Since the is a discrete distribution, it’s possible to define it using its probability measure:

where is the measure of Dirac at point [12].

3.4. Expectation

We denote:

The Esperance of the (p+1)-nomial distribution is:

Proof. We proceed as follows:

After simplifications:

We can easily deduce that:

For , we have and . We find : Esperance of binomial distribution.

For , and . We find : Esperance of trinomial distribution.

3.5. Variance

Theorem. For , the variance of the distribution is given by the formula below:

Where is a polynomial of degree .

Proof. First, calculate the Esperance of :

By applying the variance definition formula:

Let consider the following polynomial:

We have:

This means that is divisible by and not divisible by . We therefore, let factor the polynomial as follows:

Thus, the variance of the distribution becomes:

We find the expression of the polynomial using the following relation linking it to the polynomial :

3.6. Some Properties of Qp and P2p-2

The purpose of this part is to simplify the determination of the polynomials and so that the establishment of the characteristics of the p-nomial law are simple as much as possible. For this, we will proceed to factorization of the polynomial

3.6.1. First Values of p

3.6.2. Degree of Polynomial Qp

It's easy to notice that . If we note , then:

So, we can deduce that:

3.6.3. Factorization of the Polynomial Qp

We know, using the definition of the distribution, that the polynomial has no roots on . So, let's look for the complex roots of :

In addition, we have:

By discussing the cases, we’ll have,

we can summarize the two cases (for p odd or even) by the following formula:

We recall that: . Now let's look for the dominant coefficient of the polynomial . Using the relationship between the coefficients of a polynomial and its roots, we find :

We use these two results in this article proved in [10]:

We get:

if we define the staircase function of by:

We get:

We can thus write the factorization of the polynomial :

3.6.4. Rational Fraction of Variance Fp

We define the rational fraction appearing in the expression of the variance of the distribution, and we call it rational fraction of variance of order p, by:

We represent in the following figure the rational fractions of variance of order 1 to 5 for :

Figure 2. Curves of rational fractions of variance for p = 1 to 5

We notice that:

Property. We have the following property:

Proof. Let p be a non-zero natural integer.

By integration by part, we find that:

Which proves that:

Or by adding p on both sides:

So, we get:

3.7. Ordinary Moments

The ordinary moments [9] of the distribution are obtained by the recurrence relation:

Proof. Starting from the ordinary moment of order r + 1:

After simplifications, we get the desired formula.

3.8. Distribution Function

The distribution function of random variable following a distribution is given by:

3.9. Characteristic Function

The characteristic function of a random variable following a distribution is given by:

3.10. Moments Generating Function

The moments generating function [9] of a random variable X following a distribution is given by:

3.11. Cumulants Generating Function

We directly deduce the generating function of cumulants:

The generating function of factorial cumulants :

3.12. Beta-(p+1)-nomial Distribution

Similarly, we can define the distribution :

Where is the Euler’s beta function.

3.13. Markov’s Inequality

The Markov’s inequality applied for a random variable following a distribution give :

3.14. Bienaymé-Tchebychev's Inequality

The Bienaymé-Tchebychev’s inequality [8] for a random variable following a nomial distribution is obtained thanks to the moments:

3.15. Central Limit Theorem

Let a series of independent random variables of the same distribution and . The application of the central limit theorem [8,9] gives, :

4. Application

In this example, we are interested in the case of a pandemic. It is assumed that a population P of individuals is affected by a contaminating pandemic. We subdivide this population into groups of the same number of individuals . We will therefore realize trials so that for each trail we will realize screening tests by taking an individual from each group, and this in order to isolate the infected individuals.

If the result of the test is that the individual is infected, we say that it is positive and we write P, if not, we say that it is negative and we write N.

Figure 3. Subdivision of the population into p groups of n individuals

We assume that the rate of infected in the groups is the same and it’s equal to .

Detection of infected follow distribution:

We consider a simple example. Let a population of 16 individuals. We are dealing with three cases:

1- We realize 16 tests on the entire population without subdividing it into groups. This is the case with binomial distribution.

2- W subdivide this population into two groups and we’ll realize 8 tests. This is the case of trinomial distribution.

3- We subdivide this population into four groups and we’ll realize 4 tests. This is the case of pentanomial distribution.

We will study the three cases for the infection rate .

a) First case: binomial distribution

The distribution of detection of infected is:

Figure 4. Curves of detection infected in case of binomial distribution

b) Second case: trinomial distribution

The staircase of the trinomial coefficients for to is as follows:

Table

The distribution of detection of infected is:

Figure 5. Curves of detection infected in case of trinomial distribution

c) Third case: pentanomial distribution

The staircase of the pentanomial coefficients for to is as follows:

Table

The distribution of detection of infected is:

Figure 6. Curves of detection infected in case of pentanomial distribution

Note. We observe that as long as we use a p-nomial distribution of parameter p large as long as the distribution of probabilities flattens out. In other words, the peak of curves softens. This shows the usefulness of dividing the population into several groups in order to reduce the spread and infection by the pandemic.

5. Trinomial Distribution

5.1. Definition

We consider two sets and containing the same elements denoted with cardinal :

We are interested in the number of possible choices of elements among the elements that we have without repetition or also in the number of sets that we can construct from sets and such as and for .

5.2. Theorem

This number of partitions is exactly the trinomial coefficient .

Proof. The idea of demonstration of this theorem consists in considering a set of cardinal such that of the set then completing the elements which remain of the same set of the set . By gradually varying , we will get all the possible sets of cardinal :

It is quite clear from the above diagrams that:

This means that: . Furthermore:

Now the choice of elements among is and the choice of elements among is . Thus, by following this reasoning approach, the number of construction of sets of cardinal for a given is .

Finally, the total number of construction of sets of cardinal of the two sets and is:

Note. In the same way, by considering identical sets of cardinal each, we can demonstrate that the number of choices of element among the elements that we have or even the number of construction of sets of cardinal is exactly the nomial coefficient .

5.3. Probability Tree of the Trinomial Distribution

We consider two identical coins noted and . We will launch these two coins at the same time for independent trials. Each outcome has a fixed probability, the same from trial to trial.

If we get a head, we note a success S and if we get a tail, we note a failure F. The possible outcomes are , or . We note the kth trial. We represent below the probability tree:

Figure 7. Probability tree of the trinomial distribution

We count, after n tests, the total number of success obtained. If we denote by this random variable then the possible values of are . In addition, if we note the probability of having a success, then the random variable X follows a trinomial distribution noted such that :

We note:

5.4. Examples

5.4.1. Case where n = 1

In this case, one trial is realized. The possible outcomes to obtain are: , or . If we denote by the number of obtaining successes, then:

The probabilities are thus:

For :

5.4.2. Case where n = 2

In this case, two trials are realized. The possible outcomes to obtain are: , , , or . If we denote by the number of obtaining successes, then:

The probabilities are thus:

For , we get:

6. Conclusion

This article is an introduction for constructing of a new probability distribution called p-nomial distribution as a generalization of binomial distribution. The examples of application of this new probability law introduced in this research give us some ideas about its frequent meeting in practice.

References

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[2]   | Anders Hald, A History of Probability and Statistics and Their Applications before 1750, John Wiley & Sons, 2005.
[3]   | Eric Gossett, Discrete Mathematics with Proof, John Wiley & Sons, 2009.
[4]   | Michiel Hazewinkel, Encyclopaedia of Mathematics, vol. 2, Springer Science+Business Media, 1994.
[5]   | Norman Johnson, Adrienne Kemp and Samuel Kotz, Univariate Discrete Distributions, John Wiley & Sons, 2005
[6]   | Alan Ruegg, Probabilités et statistique, vol. 3, PPUR, 1994.
[7]   | Aziz ATTA, “p-nomial Coefficients and p-nomial Theorem.” Turkish Journal of Analysis and Number Theory, vol. 8, no. 1 (2020): 6-15.
[8]   | F.M. Dekking C. Kraaikamp H.P. Lopuhaa L.E. Meester, A Modern Introduction to Probability and Statistics, Understanding Why and How, Delft Institute of Applied Mathematics Delft University of Technology Mekelweg 4 2628 CD Delft The Netherlands.
[9]   | Charles M. Grinstead, J. Laurie Snell, Introduction to Probability, Swarthmore College and Dartmouth College, USA.
[10]   | Aziz ATTA, “Twin Polynomials and Kernels Matrix.” Turkish Journal of Analysis and Number Theory, vol. 8, no. 3 (2020): 57-69.
[11]   | Andreas N. Philippou, Demetrios L. Antzoulakos, Binomial Distribution, Chapter · January 2011,
[12]   | Matthew Aldridge, Measure Theory and Integration: Measures, MA40042, UK.
[13]   | M. Samuel Fiorini, MATH-F-307 Mathématiques discrètes, Version du 5 octobre 2012.