June 13, 2026
In this paper, we focus on de Moivre random experience which allows us to introduce the \(s-\)Bernoulli distribution and the bi\(^s\)nomial distribution. We present some probabilistic properties such as the expectation, the variance, the skewness and kurtosis coefficients, the moments and the generating functions. Then we establish that for \(s\in\mathbb{N}\), the bi\(^s\)nomial distribution converges to a limiting Poisson and normal distributions when \(n\rightarrow\infty.\)
2020 Mathematics Subject Classification. 60C05, 60E05, 11B65, 11B68, 05A15, 05A10.
Keywords. The de Moivre experiment , bi\(^s\)nomial coefficients , \(s-\)Bernoulli distribution , bi\(^s\)nomial distribution , moments function , Eulerian numbers , Bernoulli numbers , partial Bell polynomials , Poisson distribution , normal distribution polynomials.
In 1711, Abraham de Moivre discussed in his book [1] through different examples, the bi\(^s\)nomial coefficients. The main result appeared in a lemma which stated: "To find how many chances there are upon any number of dice, each of them of the same given number of faces, to throw any given number of points" [1]. If we denote by \(n\) the number of dices, \((s+1)\) the number of faces and \(k\) the number of points, we denote this number by \({n \choose k}_s.\) Several extensions and studies have been investigated in the literature, for example Bondarenko [2] gave a combinatorial interpretation of bi\(^s\)nomial coefficients \({n\choose k}_s\) as the number of different ways of distributing \(k\) balls among \(n\) cells where each cell contains at most \(s\) balls. If we denote \(x_i\) the number of balls in a cell, the interpretation given by Bondarenko is equivalent to evaluate the number of solutions of the system \[\left\lbrace \begin{array}{ll} x_1+\cdots+x_n=k,\\ 0\leq x_1,\ldots,x_n\leq s. \end{array}\right. \label{eq321}\tag{1}\]
Belbachir in [3] got an explicit relation with one summation symbol to compute the bi\(^{s}nomial\) coefficients.
\[{n\choose k}_s = \sum_{j=0}^{\lfloor k/(s+1)\rfloor}(-1)^j {n\choose j} {k-j(s+1)+n-1\choose n-1}. \label{eq322}\tag{2}\]
In fact, de Moivre see [1] solves the system 1 by giving the right hand side of 2 .
This formula is an alternate expression, which is difficult to handle in probability, we prefer expressions where there are only positive terms. Let’s start from this problem which interested the community at that time but could not be modelized since the simplified expressions were only given during the 2000s.
In the present work, the random experiment on which we will build our distribution is de Moivre’s experiment, [1], where he stated the following probability problem “there are \(n\) dices with \((s + 1)\) faces, if these are thrown randomly, what would be the chance of the exhibited sum of numbers to be equal to \(k\)?”;
Knowing that at the time we could not modelize; it is a probability distribution that has never been considered in the literature. It was considered when the number of faces equals to 2, i.e. the coin toss problem, with which we start Bernoulli’s and the binomial’s distributions. So it is the modelization of a uniform distribution on \(\left\lbrace 0,1\right\rbrace\), but the modelization of the uniform distribution on \(\left\lbrace 0,\ldots,s\right\rbrace\) has rarely been developed in the literature. We propose in this paper to develop it as well as the distribution laws associated to it.
The experiment consists of tossing a die with \((s+1)\) faces numbered in \(\{0,1,\ldots,s\}\). Note that \(p\) and \(q\) are the probability of success and failure respectively, the scheme of the different possible outcomes is as follows : \[\begin{array}{|c|c|c|c|c|c|} \hline Face ( k )& 0 & 1 & 2&\cdots&s\\ \hline \text{Probability }(p_k)& q^s & p q^{s-1} & p^2q^{s-2}&\cdots&p^s\\ \hline \end{array}\]
The bi\(^s\)nomial distribution, like the classical binomial distribution, can converge to a limiting the Poisson and normal distribution, where in this work we propose the proof of this convergence.
The Poisson distribution was introduced in 1837 by Denis Poisson, in his work Recherches sur la probabilité des jugements en matière criminelle et en matière civile [4]. In particular, in Chapter 8, he obtained his distribution as the limit of the binomial distribution \(B (T, \lambda / T)\) when \(T\) becomes large, but moved on fairly quickly to other considerations. He does not study this limit distribution. In 1711, de Moivre had obtained the same distribution in [5] still as the limit of the Binomial distribution. This has prompted some authors in [6], [7], to argue that the Poisson distribution should bear the name of de Moivre.
On the other hand, the first appearance of a normal distribution was in 1733 by Abraham de Moivre who by deepening the study of the factorial \(n!\) when studying a coin toss game. He published in The doctrine of chances [8], in which a normal distribution appears as the limit of a binomial distribution, which will be at the origin of the central limit theorem. In 1777, Pierre-Simon de Laplace resumes this work and obtains a good approximation of the error between this normal distribution and a binomial distribution using Euler’s gamma function [9].
A normal distribution is then fully defined when the first central limit theorem, then called Laplace’s theorem, is stated by Laplace in 1812 [9]. Its name “normal” is given by Henri Poincaré at the end of the nineteenth century [6].
In what follows, after having presented the de Moivre experiment, we define in the next sections the \(s\)-Bernoulli and bi\(^s\)nomial distributions respectively with their properties, then in the fifth and sixth sections we introduce the \(s-\)Poisson and the \(s-\)normal distributions respectively as a limiting case of the bi\(^s\)nomial distribution in the case where \(p\neq q\).
The coefficients \({ n \choose k}_s\) were introduced first by the work of Abraham de Moivre [1] and later, Euler [10], [11], studied these coefficients and derived a number of properties, some results were developed by Euler using the construction of the \(s-\)Pascal triangle which is a natural generalization of Pascal triangle. In 1876, André [12], studied these coefficients using combinations on words and explored several properties, he gave the absorption identity of bi\(^s\)nomial coefficients [12], which was generalized by Belbachir and Igueroufa see [13].
These coefficients have appeared in history under different names. We can cite for example, generalized binomial coefficients in 1993 by Bondarenko [2], ordinary multinomial coefficients in 1997 by Ericksen [14] and Warnaar [15]. In our work we use the bi\(^s\)nomial coefficients name which was introduced in [16].
Using the classical binomial coefficient, one has de Moivre alternate summation [5] \[{n \choose k}_s =\sum_{j=0}^{\lfloor k/(s+1) \rfloor}(-1)^j {n\choose j} {k-j(s+1)+n-1\choose n-1}.\]
We define a random variable \(X\) taking its values in \(X(\Omega)= \{0,1,2,\ldots,s\}\), we call \(X\) a random variable following the \(s\)-Bernoulli distribution of parameter \(p\) and denoted by \(X\rightsquigarrow \mathcal{B}^{(s)}(p)\). We define the probability distribution function of \(X\) as follows :
The probability distribution is given by \[\mathit{P}(X=k)=\left\{{\begin{array}{llll}p^s&\quad {for }k=s,\\p^{s-1}q&\quad {for }k=s-1,\\ \vdots\\q^s&\quad {for } k=0,\end{array}}\right. \label{eq:7}\tag{3}\] with \(\sum_{k=0}^{s}{\mathit {P}}(X=k)=1\) or
\[\frac{p^{s+1}-q^{s+1}}{p-q} =1 \text{ equivalent to } p\left( p^s-1\right) =q\left( q^s-1\right), \text{ for } p\neq q. \label{eq7}\tag{4}\]
\[p = (s+1)^{-1/s}, \text{ for } p=q.\]
In the following theorem, we will present the generating function of the \(s\)-bernoulli distribution.
Theorem 1. Let \(X\) be a random variable that follows an \(s\)-Bernoulli distribution of probability \(p\), \(X \rightsquigarrow \mathcal{B}^{(s)} (p)\), then the generating function of an \(s\)-Bernoulli distribution is, for \(p\neq q\) \[G_X(t)=E(t^X) = \sum_{k=0}^{s} t^k p^k q^{s-k} =\frac{(pt)^{s+1}-q^{s+1}}{pt-q},\label{eq:5}\tag{5}\]
\[G_X(t)=\frac{1}{s+1}\left(\frac{1-t^{s+1}}{1-t}\right), \qquad\text{ for } p=q. \label{eq32:325461}\tag{6}\]
Corollary 1. Let \(X\) be a random variable that follows an \(s\)-Bernoulli distribution of probability \(p\), \(X \rightsquigarrow \mathcal{B}^{(s)} (p)\), then the expectation and the variance of \(X\) are \[E(X) = \frac{p \left((s+1) p^{s}-1\right)}{(p-q)},\qquad V(X) =\frac{p q\left( 1-(s+1)^2 (p q)^s\right) }{(p-q)^2}, \text{ for } p\neq q \label{eq:6}\tag{7}\]
\[E(X)= s/2,\qquad V(X)=s(s+2)/12, \qquad \text{ for }p=q.\]
Proof. Using the generating function given in 5 . The expectation of \(X\), \(E(X)\), when \(p\neq q\), is given by \[G'_X(1)=E(X)= \frac{(s+1) p^{s+1}}{p-q}-\frac{p \left(p^{s+1}-q^{s+1}\right)}{(p-q)^2}.\] It is a direct computation from 5 and 6 . ◻
In order to introduce higher moments order, we need to consider the moment-generating function for the s-Bernoulli distribution,
\[M_X(t)=E(e^{tX})=\frac{(p e^t)^{s+1}-q^{s+1}}{pe^t-q}.\]
The explicit expression of this function showed a relation with the explicit formula of the Eulerian numbers, [17], given by
\[A(n,k) =\sum _{j=0}^k (-1)^{j} \binom{n+1}{j} (k+1-j)^n,\qquad \;0\leq k\leq n-1,\label{eq19}\tag{8}\] where \(A(n,k)\) are called Eulerian
numbers.
By introducing the parameter \(s\), we get an extended Eulerian number form.
Definition 1. For integers \(n\), \(k\) and \(s\), the extended Eulerian number is defined by \[A(n,k,s) = \sum _{j=0}^k (-1)^{j} \binom{n+1}{j} (k+1-j-s)^n, \qquad \;0\leq k\leq n. \label{eq3223}\tag{9}\] For \(s=0\) the summand reduces to that of 8 , so we recover the classical Eulerian numbers on their support, \(A(n,k,0)=A(n,k)\) for \(0\leq k\leq n-1\), with the additional value \(A(n,n,0)=0\).
Theorem 2. The function of the moments of the s-Bernoulli distribution is given by,
\[m_r(s) =\frac{(-1)^r }{(p-q)^{r+1}} p \sum_{k=0}^r\left( A(r,k,s+1) p^{r-k+s} q^{k} - A(r,k,1) p^{r-k} q^{k+s} \right),\quad\text{ for } p\neq q,\label{eq:mr}\tag{10}\]
\[m_r(s) = \frac{1}{(r+1)(s+1)}\sum_{k=0}^r(-1)^k { r+1 \choose k} B_k s^{r-k+1} , \quad\text{ for } p= q,\] the constants \(B_k\), for \(p=q\), are the Bernoulli numbers, see [18].
Proof. For \(p\neq q\), the successive derivatives of the moment-generating function generate extended Eulerian numbers \(A(r,k,s)\). We obtain in the numerator two polynomial forms. The first is generated by the coefficients \(A(r,k,s+1)\) and the second by the extended Eulerian numbers \(A(r,k,1)\). For \(p= q\), we find a Faulhaber’s like formula \(m_r(s)=p^s\sum_{k=1}^s k^r\) which can be expressed in terms of the Bernoulli numbers. ◻
We denote by \(\mu_r=E(X-E(X))^r\) and \(\sigma=\sqrt{\mu_2} .\) Then the skewness and the kurtosis are denoted respectively by \(\gamma_1=\mu_3/\sigma^3\) and \(\gamma_2=\mu_4/\sigma^4 .\)
We take as an example, the third and the fourth moments for any \(s\), when \(p=q\), which are respectively \[m_3(s)=\frac{1}{4} s^2 (s+1), \qquad m_4(s)=\frac{1}{30} s \left(6 s^3+9 s^2+s-1\right),\] where the skewness is null and the kurtosis is as follows \[\gamma_2=\frac{3 \left(3 s^2+6s-4\right)}{5 s (s+2)}.\]
The bi\(^ s\)nominal distribution is obtained by repeating \(n\) times, the de Moivre experiment, identically and independently of each other. Or instead of a single die we take \(n\) dice with \(s+1\) faces, we observe a random variable \(X\) which is equal to the number of \(s\) successes taking its values between 0 and \(sn\). This random variable has the distribution probability as the bi\(^ s\)nomial distribution of parameters \((n, p)\).
We define a random variable \(X\) taking its values in the set \(\{0,1,2, \ldots, sn \}\), we call \(X\) a variable following the bi\(^s\)nomial distribution of parameters \((n, p)\) as the discrete probability distribution of a random variable \(X\) denoted by \(X\leadsto \mathcal{B}^{(s)}(n,p)\) which has the mass function given by : \[\mathbb{P}(X=k)= { n \choose k}_s p^kq^{sn-k},\quad 0\leq k\leq sn, \label{26}\tag{11}\] under the condition \(\sum_{k=0}^{s} p^k q^{s-k}=1\).
For \(s=1\), we find the classical binomial probability mass function.
For \(p=q\), we have the uniform case : The \(s\)-Uniform Distribution
\[P(X=k)= \large\frac{{n \choose k}_s}{(s+1)^{n}}.\]
In [19], Belbachir specified the smallest mode of the bi\(^s\)nomial distribution leading to the expression of the maximal probability is given by this formula \[k_n:=\arg\underset{k}{\max}{n\choose k}_s.\]
In the following theorem, we will present the generating function of the bi\(^s\)nomial distribution.
Theorem 3. Let \(X\) be a discrete random variable with the bi\(^s\)nomial distribution with the parameters \(n\) and \(p\), \(X\leadsto \mathcal{B}^{(s)}(n,p)\), then the generating function is \[G_X(t) =\left(\frac{q^{s+1}-(pt)^{s+1} }{q-p t}\right)^n=\sum_{k=0}^{sn}{n\choose k}_s (pt)^kq^{sn-k},\quad \text{ for } p\neq q,\label{eq:19}\tag{12}\]
\[G_X(t) =\frac{1}{\left( s+1\right)^n}\left( \frac{1-t^{s+1}}{1-t}\right)^n=\frac{1}{\left( s+1\right)^n}\sum_{k=0}^{sn}{n\choose k}_s t^k, \quad \text{ for } p=q.\]
Corollary 2. Let \(X\) be a discrete random variable with the bi\(^s\)nomial distribution with the parameters \(n\) and \(p\), \(X\leadsto \mathcal{B}^{(s)}(n,p)\), the expectation and the variance are given by \[E(X) =np\frac{(s+1)p^{s}-1}{(p-q)},\qquad V(X) =npq\frac{ 1-(s+1)^2(pq)^s}{(p-q)^2}, \quad \text{ for } p\neq q,\]
\[E(X) =sn/2, \qquad V(X) = sn(s+2)/12,\quad \text{ for } p= q.\]
Proof. Using the generating function given in 12 . We can show that the expectation of \(X\), \(E(X)\), it suffices to use the derivatives of the generating function. ◻
The calculation of higher moments for the bi\(^s\)nomial distribution shows a connection to some special case of the alternate sum of partial Bell polynomials of \(x_1, x_2,\ldots\),
defined by their generating function as follows, see [17] \[\sum_{n=k}^{\infty}B_{n,k}(x_1,x_2,\ldots)\frac{t^k}{n!}=\frac{1}{k!}\left(
\sum_{m=1}^{\infty}x_m \frac{t^m}{m!}\right)^k.\] The explicit formula for the \(B_{r,k}(x_1,x_2,\ldots)\;\) is given by \[B_{n,k}(x_{1},x_{2},\ldots
)=\sum_{\substack{ j_{1}+j_{2}+\cdots =k\\ j_{1}+2j_{2}+\cdots=n}} {n! \over j_{1}!j_{2}!\cdots }\left({x_{1} \over 1!}\right)^{j_{1}} \left({x_{2} \over 2!}\right)^{j_{2}}\cdots,\label{31}\tag{13}\] We note that the formula 13 admits a finite number of terms according to \(j_{1}+2j_{2}+3j_3+\cdots=n .\) Thereafter we can use one of the two notations \(B_{n,k}(x_{1},x_{2},\ldots )\) or \(B_{n,k}(x_{1},x_{2},\ldots,x_{n-k+1} )\), for more details we can refer to [17].
Theorem 4. The bi\(^{s}\)nomial distribution moments
The bi\(^{s}\)nomial distribution moments of the order \(r\), denoted by \(M(r)\), has a polynomial form that is generated by the partial Bell polynomials.
\[M(r)=\sum_{k=1}^{r} (n)_k B_{r,k}(\sum_{l=1}^{s}l p^l q^{s-l},\sum_{l=1}^{s}l^2 p^l q^{s-l},\ldots,\sum_{l=1}^{s}l^{r-k+1} p^l q^{s-l}), \quad \text{ for } p\neq q,\]
\[M(r)=\sum_{k=1}^{r} (n)_k p^{(k s)} B_{r,k}(\sum_{l=1}^{s}l ,\sum_{l=1}^{s}l^2,\ldots,\sum_{l=1}^{s}l^{r-k+1} ),\quad \text{ for } p= q.\] where \((n)_k\) is the falling factorial, \((n)_k=n(n-1)\cdots(n-k+1).\)
Proof. By taking \(x_i = \sum_{l=1}^{s} l^{i}p^l q^{(s-l)}\) in the partial Bell polynomials, multiplied by the \(kth\) falling factorial power of \(n\) and summing on \(k\), then we get the expression of the bi\(^ s\)nomial distribution moments. On the other hand taking \(x_i = p^{s}\sum_{l=1}^{s} l^{i}\) we get the the \(r^{th}\) moment in the case of \(p=q\). ◻
For \(p=q\), the third and fourth moments are \[\begin{align} M(3)&= n^2 s^2 (n s+s+2)/8,\\ M(4)&=sn\left( 15 n^3 s^3+30 n^2 s^2 (s+2)+5 sn (s+2)^2-2 (s^3+4s^2+6s+4)\right)/240. \end{align}\]
Also the skewness is null and the kurtosis is as follows \[\gamma_2=\frac{3((5 n-2) s (s+2)-4)}{5sn (s+2)}.\]
Classically, the Poisson distribution is a limiting case of the binomial distribution which arises when the number of trials \(n\) increases indefinitely while the parameter \(\lambda\), which is the expected value of the number of successes from the trials, remains constant. Using the same approach we do the same for the bi\(^s\)nomial distribution.
Lemma 1. Let \(k\) and \(n\) be integers, we have \[\lim_{n\rightarrow\infty}\dfrac{{n\choose k}}{n^k}=\dfrac{1}{k!}\]
Proof. We have \({n\choose k}=\frac{n(n-1)\cdots(n-k+1)}{k!}\), if we take the limit
\(\lim\limits_{n\rightarrow\infty}\dfrac{{n\choose k}}{n^k}=\frac{n(n-1)\cdots(n-k+1)}{k! n^k}=\dfrac{1}{k!} .\) ◻
Lemma 2. Let \(s,k\) and \(n\) be integers, we have \[\lim\limits_{n\rightarrow\infty} \dfrac{{n \choose k}_s}{n^k}=\dfrac{1}{k!}.\]
Proof. To find the limit \(\lim\limits_{n\rightarrow\infty} \dfrac{{n \choose k}_s}{n^k}\). We use the formula of Belbachir and Benmezai in [16].
\[{n \choose k }_s =(-1)^k \sum\limits_{j_1+j_2+\cdots+j_s=k} {n \choose j_1} {n \choose j_2}\cdots {n \choose j_s} a^{j_1+2j_2+\cdots+sj_s},\] with \(a=\exp\left( -2\pi i/(s+1)\right)\), where \(i^2=-1.\)
\[\begin{align} \lim\limits_{n\rightarrow +\infty} \frac{{ n \choose k}_s}{n^k}&= (-1)^k \sum\limits_{j_1+\cdots+j_s=k} \lim\limits_{n\rightarrow +\infty}\frac{{n \choose j_1}}{n^{j_1}} \frac{{n \choose j_2}}{n^{j_2}}\cdots \frac{{n \choose j_s}}{n^{j_s}} a^{j_1+2j_2+\cdots+sj_s} \\&=(-1)^k \sum\limits_{j_1+\cdots+j_s=k} \frac{1}{j_1!} \frac{1}{j_2!}\cdots \frac{1}{j_s!} a^{j_1+2j_2+\cdots+sj_s} \\&= \frac{(-1)^k}{k!} \sum\limits_{j_1+\cdots+j_s=k} {k \choose j_1,\cdots,j_s}a^{j_1+2j_2+\cdots+sj_s}\\&= \frac{(-1)^k}{k!}(a+a^2+\cdots+a^s)^k\\&= \frac{(-1)^k}{k!} \left( \frac{a-a^{s+1}}{1-a}\right) ^k,\\ \end{align}\] since \(a^{s+1}=\exp\left( -2\pi i\right) =1\) then \[\lim\limits_{n\rightarrow +\infty} \frac{{ n \choose k}_s}{n^k}=\frac{(-1)^k}{k!} (-1)^k =\frac{1}{k!}.\] ◻
Theorem 5. Let \(s\) be a non-zero integer. Let \((p_n)_n\) and \((q_n)_n\) be two sequences of real numbers in the interval \(]0,1[\). Let \((X_n)_n\) be a sequence of random variables, where each \(X_n\) takes its values in the set \(\{0, 1, \dots,sn\}\) and follows a bi\(^s\)nomial distribution \(\mathcal{B}^{(s)}(n, p_n)\). Assume the sequence of expectations converges to a finite, positive limit \(\lambda_s\), such that: \[\lim_{n \to \infty} E(X_n) = \lim_{n \to \infty} np_n \frac{(s+1)p_n^s - 1}{p_n - q_n} = \lambda_s > 0,\] subject to the condition for each \(n\): \[p_n^s + p_n^{s-1}q_n + \cdots + q_n^s = 1.\] Then, the sequence of random variables \((X_n)_n\) converges in law to a Poisson distribution with parameter \(\lambda_s\). In other words, for any integer \(k \ge 0\): \[\lim_{n \to \infty} P(X_n=k) = \lim_{n \to \infty} {n \choose k}_s p_n^k q_n^{sn-k} = \frac{\lambda_s^k}{k!}e^{-\lambda_s}.\]
Proof. The main goal is to evaluate the following limit \[\lim\limits_{n\rightarrow\infty} {n \choose k}_s p_n^k q_n^{sn-k}, \text{ with } p_n^s+p_n^{s-1}q_n+\cdots+q_n^s=1. \label{eq:2}\tag{14}\] We should use this system to get \(p_n\) and \(q_n\) in terms of \(\lambda_s\). \[\left\lbrace \begin{array}{ll} \sum\limits_{k=0}^{sn}P(X_n=k)=1,\\ E(X_n)=\lambda_s. \end{array} \right. \Leftrightarrow \left\lbrace \begin{array}{ll} p_n^{s+1} -q_n^{s+1}-p_n+q_n=0,\\ np_n((s+1)p_n^{s}-1)-(p_n-q_n)\lambda_s=0 . \end{array} \right. \label{eq3213}\tag{15}\]
In a first step, from the first equation of the system 15 , we apply the Implicit Function Theorem to get an approximation of \(q_n\) in terms of \(p_n\) at the order two which is large as enough for our proof.
We give the statement of the theorem and we can find the proof in [20].
Theorem 6. [Implicit Function Theorem][20] Assume that \(\Omega\) is an open subset of \(\mathbb{R}^2\) and that \(f : \Omega\rightarrow \mathbb{R}\) is a function of class \(C^k\). Assume also that \((a,b)\) is a point in \(\Omega\) such that \[f(a,b)=0 \qquad\qquad \text{ and }\qquad\qquad \frac{\partial f}{\partial y}(a,b)\neq 0.\] Then there exists an open set neighborhood \(U\) of \((a,b)\) in \(\mathbb{R}^2\), an open interval \(I\) of \(\mathbb{R}\) which contains \(a\), and a function \(g:I\rightarrow\mathbb{R}\) of class \(C^k\) such that, for any \((x,y)\in U\) we have \[f(x,y)=0\Leftrightarrow y=g(x).\] Moreover, for all \(x \in I\), we have \[g'(x)=-\dfrac{\dfrac{\partial f}{\partial x}(x,g(x))}{\dfrac{\partial f}{\partial y}(x,g(x))}.\]
Let \(f\) be a function defined by \(f(p_n,q_n)=p_n^{s+1} -q_n^{s+1}-p_n+q_n\). We have \[\dfrac{\partial f(p_n,q_n)}{\partial q_n}=-(s+1)q_n^{s}+1\Rightarrow \dfrac{\partial f(0,1)}{\partial q_n}=-s\neq0.\] Since in addition \(f(0,1)=0\), by applying Theorem 6. There are two open intervals \(I\) and \(J\), with \(0\in I\) and \(1\in J\), and a function \(g:I\rightarrow J\) of class \(C^1\) such that \[\text{ for all }\qquad(p_n,q_n)\in I \times J, f(p_n,q_n)=0\Leftrightarrow q_n=g(p_n).\]
Moreover, since \(f\) is of class \(C^\infty\), \(g\) is also of class \(C^\infty\) on \(I\). In particular, there exists a power series of any order around 0, that of order two is written
\[q_n= g(0)+\dfrac{g'(0)}{1!}p_n+\dfrac{g''(0)}{2!}p_n^2+o(p_n^2). \label{eq14}\tag{16}\]
We already know that \(g(0)=1\). To calculate \(g'(0)\) and \(g''(0)\), we derive twice the function
\[f\left(p_n,g(p_n)\right) = p_n^{s+1} -g(p_n)^{s+1}-p_n+g(p_n).\]
We derive a first time and evaluate for \(p_n=0\), we find
\[(s+1)p_n^s-(s+1)g'(p_n)g(p_n)^{s}-1+g'(p_n)=0\Rightarrow g'(0)=\dfrac{-1}{s}.\]
We derive a second time and evaluate for \(p_n=0\) to find
\[\begin{align} & (s+1)sp_n^{s-1}-(s+1) g''(p_n)g(p_n)^{s}-(s+1)s g'(p_n)^2g(p_n)^{s-1}+g''(p_n)=0\\&\Rightarrow g''(0)=-\dfrac{s+1}{s^2}. \end{align}\]
Thus, by substituting in 16 , we get
\[q_n=1-\frac{p_n}{s}-\frac{(s+1)p_n^2}{2s^2}+o(p_n^2). \label{eq:4}\tag{17}\]
In a second step, we give \(p_n\) in terms of \(\lambda_s\) from the second equation of the system 15 . In the same way, we suppose a function \(f\) defined by \[f(p_n,\lambda_s)=np_n((s+1)p_n^{s}-1)+\left( 1-\frac{s+1}{s}p_n-\frac{(s+1)p_n^2}{2s^2}+o(p_n^2)\right) \lambda_s.\]
We have \[\begin{align} &\dfrac{\partial f(p_n,\lambda_s)}{\partial p_n}=n((s+1)p_n^{s}-1)+n(s+1)sp_n^s-\left( \frac{s+1}{s}+\frac{(s+1)}{s^2}p_n+o(p_n)\right) \lambda_s\\&\Rightarrow \dfrac{\partial f(0,0)}{\partial p_n}=-n\neq0. \end{align}\] Since in addition \(f(0,0)=0\), we apply the Implicit Function Theorem, we define a function \(h:I\rightarrow J\) of class \(C^1\) such that \(\text{ for }(p_n,\lambda_s)\in I \times J\) \[f(p_n,\lambda_s)=0\Leftrightarrow p_n=h(\lambda_s).\]
So there exists a power series of any order around 0, the one of order two is given by
\[p_n= h(0)+\dfrac{h'(0)}{1!}\lambda_s+\dfrac{h''(0)}{2!}\lambda_s^2+o(\lambda_s^2). \label{eq16}\tag{18}\]
We already know that \(h(0)=0\). To calculate \(h'(0)\) and \(h''(0)\), we derive twice the function \[f(h(\lambda_s),\lambda_s) =nh(\lambda_s)((s+1)h(\lambda_s)^{s}-1)+(1-\frac{s+1}{s}h(\lambda_s)-\frac{(s+1)h(\lambda_s)^2}{2s^2}+o(h(\lambda_s)^2))\lambda_s.\]
We derive over \(\lambda_s\) a first time and evaluate for \(\lambda_s=0\) we find
\[\begin{align} & nh'(\lambda_s)((s+1)h(\lambda_s)^{s}-1)+n(s+1)sh'(\lambda_s)h(\lambda_s)^s+\\&\left(1- \frac{s+1}{s}h(\lambda_s)-\frac{(s+1)}{s^2}h(\lambda_s)^2+o(h(\lambda_s)^2)\right)-\\&\left( \frac{s+1}{s}h'(\lambda_s)+\frac{(s+1)}{s^2}h'(\lambda_s)h(\lambda_s)+o(h(\lambda_s))\right) \lambda_s=0 \\& \text{ which gives }h'(0)=\dfrac{1}{n}. \end{align}\] We derive over \(\lambda_s\) a second time and evaluate for \(\lambda_s=0\) to find \[\begin{align} &n h''(\lambda_s ) \left((s+1) h(\lambda_s )^s-1\right)+2 n s (s+1) h'(\lambda_s )^2 h(\lambda_s )^{s-1}+\\&n (s+1) h(\lambda_s ) \left(s h''(\lambda_s ) h(\lambda_s )^{s-1}+(s-1) s h'(\lambda_s )^2 h(\lambda_s )^{s-2}\right)-\\&2 \left(\frac{(s+1) h(\lambda_s ) h'(\lambda_s )}{s^2}+\frac{(s+1) h'(\lambda_s )}{s}\right)-\\& \left(\frac{(s+1) h''(\lambda_s )}{s}+\frac{(s+1) \left(2 h(\lambda_s ) h''(\lambda_s )+2 h'(\lambda_s )^2\right)}{2 s^2}\right)\lambda_s=0\\&\Rightarrow h''(0)=-\dfrac{2(s+1)}{sn^2}. \end{align}\]
By substituting in 18 , we get
\[p_n=\frac{\lambda_s}{n}-\frac{s+1}{sn^2}\lambda_s^2+o(\lambda_s^2). \label{eq:17}\tag{19}\]
We substitute \(p_n\) in 17 to obtain the expression of \(q_n\) in terms of \(\lambda_s\), \[q_n= 1-\frac{\lambda_s}{sn}+\frac{s+1}{2(sn)^2}\lambda_s^2+o(\lambda_s^2).\]
We replace the two expressions of \(p_n\) and \(q_n\) in 14 \[\begin{align} &\lim\limits_{n\rightarrow\infty} {n \choose k}_s p_n^k q_n^{sn-k}\\&=\lim\limits_{n\rightarrow\infty} {n \choose k}_s \left(\frac{\lambda_s}{n}-\frac{s+1}{sn^2}\lambda_s^2+o(\lambda_s^2)\right)^k \left(1-\frac{\lambda_s}{sn}+\frac{s+1}{2(sn)^2}\lambda_s^2+o(\lambda_s^2)\right)^{sn-k} \\&=\lambda_s^k e^{-\lambda_s}\lim\limits_{n\rightarrow\infty} \dfrac{{n \choose k}_s}{n^k}. \end{align}\]
Then, using Lemma 2, we obtain the convergence to the Poisson distribution
\[\lim\limits_{n\rightarrow +\infty} { n \choose k}_s p_n^k q_n^{sn-k} =\frac{ \lambda_s^k}{k!} e^{-\lambda_s} .\] ◻
The Poisson distribution \(\mathcal{P}(\lambda_s)\) is the limiting distribution of the bi\(^s\)nomial distribution as \(n\) tends to infinity. We note that the expectation and the variance of the bi\(^s\)nomial distribution converge towards those of the \(s-\)Poisson distribution when we tend to infinity.
The approximation of the binomial distribution in the particular limiting case where the number of possible observations \(n\) becomes infinite and the probability of success \(p\) for each is large in this case, \(np>>1\), gives a normal distribution. We approach this binomial distribution by the normal distribution having the same expectation \(\mu\) and the same variance \(\sigma^2\). We are going to apply this concept on the bi\(^s\)nomial distribution and see how is converging to the normal distribution.
Theorem 7. Let \(X\) be a bi\(^{ s }\)nomial random variable with parameters \(n\) and \(p\), \(\mathcal{B}^s(n,p)\).
The bi\(^s\)nomial random variable \(X\) has, for large \(n\), an approximate normal distribution with mean \(\mu_s=np\frac{(s+1)p^{s}-1}{(p-q)}\) and variance \(\sigma^2_s=npq\frac{ 1-(s+1)^2(pq)^s}{(p-q)^2} .\)
Proof. The MGF method
The moment generating function of the random variable \(X\) evaluates to
\(M_X(T)=\sum\limits_{k=0}^{s n}e^{t k}{n\choose k}_s p^k q^{sn-k}=\left( \sum\limits_{k=0}^{s} \left( e^t p\right) ^k q^{s-k}\right)^n\).
Let \(Z=\frac{1}{\sigma_s}\left( X-E(X) \right)\). Below we derive the mgf of \(Z\), which is given by \[M_Z(t)=E(e^{tZ})=\left( \sum\limits_{k=0}^{s} \exp\left( \left( k-\frac{E(X)}{n}\right) \frac{t}{\sigma_s}\right) p^k q^{s-k}\right)^n\]
The Taylor series expansion for \(\exp\left( \left( k-\frac{E(X)}{n}\right) \frac{t}{\sigma_s}\right)\) gives \[\begin{align} \exp\left( \left( k-\frac{E(X)}{n}\right)
\frac{t}{\sigma_s}\right)=&1+\left( k-\frac{E(X)}{n}\right) \frac{t}{\sigma_s}+\frac{1}{2!}\left( k-\frac{E(X)}{n}\right)^2\left(\frac{t}{\sigma_s}\right)^2+\\&\frac{1}{3!}\left( k-\frac{E(X)}{n}\right)^3
\left(\frac{t}{\sigma_s}\right)^3+\frac{1}{4!}\left( k-\frac{E(X)}{n }\right)^4\left( \frac{t}{\sigma_s}\right) ^4e^{\psi(n)}, \end{align}\] where \(\psi(n)\) is a number between 0 and \(\left( k-\frac{E(X)}{n }\right) t\), and \(\psi(n)\rightarrow0\) as \(n\rightarrow\infty.\) Now substituting this equation in the last expression for \(M_Z(t)\), we have \[\begin{align} M_Z(t)&=\Biggl(\sum\limits_{k=0}^{s} p^k q^{s-k}+\sum\limits_{k=0}^{s}\left( k-\frac{E(X)}{n }\right)p^k q^{s-k} \left(
\frac{t}{\sigma_s}\right) +\frac{1}{2!}\sum\limits_{k=0}^{s}\left( k-\frac{E(X)}{n }\right)^2 p^k q^{s-k} \left( \frac{t}{\sigma_s}\right) ^2\\&+\frac{1}{3!}\sum\limits_{k=0}^{s}\left( k-\frac{E(X)}{n }\right)^3 p^k q^{s-k} \left(
\frac{t}{\sigma_s}\right) ^3+\frac{1}{4!}\sum\limits_{k=0}^{s}\left( k-\frac{E(X)}{n }\right)^4p^kq^{s-k}\left( \frac{t}{\sigma_s}\right) ^4e^{\psi(n)}\Biggr)^n, \end{align} \label{eq24}\tag{20}\] where \(\sum\limits_{k=0}^{s} p^k q^{s-k}=1\), \(\sum\limits_{k=0}^{s}\left( k-\frac{E(X)}{n }\right)p^k q^{s-k} =0\) as
\(\sum\limits_{k=0}^{s} kp^k q^{s-k}-\left( \sum\limits_{k=0}^{s}p^k q^{s-k}\right) \frac{E(X)}{n}= \sum\limits_{k=0}^{s} kp^k q^{s-k}- \frac{E(X)}{n}=\frac{E(X)}{n}- \frac{E(X)}{n}=0\),
and \(\sum\limits_{k=0}^{s}\left( k-\frac{E(X)}{n }\right)^2\left( \frac{1}{\sigma_s}\right) ^2 p^k q^{s-k} = \,\tfrac{1}{n}\) as \(\sum\limits_{k=0}^{s}\left( k-\frac{E(X)}{n}\right)^2 p^k q^{s-k}
=\frac{\sigma_s^2}{n}\), we have \[\begin{align} \sum\limits_{k=0}^{s}\left( k-\frac{E(X)}{n}\right)^2 p^k q^{s-k}&=\sum\limits_{k=0}^{s}\left( k^2-2k\frac{E(X)}{n}+\left(\frac{E(X)}{n}\right)^2\right) p^k
q^{s-k}\\&=\sum\limits_{k=0}^{s} k^2p^k q^{s-k}-2\frac{E(X)}{n}\sum\limits_{k=0}^{s}kp^k q^{s-k}+\left(\frac{E(X)}{n}\right)^2\sum\limits_{k=0}^{s}p^k
q^{s-k}\\&=\frac{E(X^2)}{n}-2\left(\frac{E(X)}{n}\right)^2+\left(\frac{E(X)}{n}\right)^2\\&=\frac{E(X^2)}{n}-\left(\frac{E(X)}{n}\right)^2\\&= \frac{\sigma_s^2}{n}. \end{align}\]
By taking \(\sigma_s^2=n\left( \sum\limits_{k=0}^{s}k^2p^kq^{s-k} -\left( \frac{E(X)}{n}\right)^2 \right)\)
Now, substituting these equations in the last expression for \(M_Z(t)\) in 20 , we have
\[\begin{align} M_Z(t)&= \Biggl(1 +\frac{t^2}{2n}+\frac{1}{3!} \left( \frac{\sum\limits_{k=0}^{s}\left( k-\frac{E(X)}{n }\right)^3p^k q^{s-k}}{n^\frac{3}{2}\left( \sum\limits_{k=0}^{s}k^2p^kq^{s-k} -\left( \frac{E(X)}{n}\right)^2 \right)^{\frac{3}{2}}}\right)t^3\\&+\frac{1}{4!}\left( \frac{\sum\limits_{k=0}^{s}\left( k-\frac{E(X)}{n }\right)^4p^k q^{s-k}}{n^2\left( \sum\limits_{k=0}^{s}k^2p^kq^{s-k} -\left( \frac{E(X)}{n}\right)^2 \right)^2}\right)t^4e^{\psi(n)}\Biggr)^n \end{align}\]
The above equation may be written as \[M_Z(t)= \Biggl(1 +\frac{t^2}{2n}+\frac{\Psi(n)}{n}\Biggr)^n,\]
where
\[\begin{align} \Psi(n)&= \frac{1}{3!} \left( \frac{\sum\limits_{k=0}^{s}\left( k-\frac{E(X)}{n }\right)^3p^k q^{s-k}}{\sqrt{n}\left( \sum\limits_{k=0}^{s}k^2p^kq^{s-k} -\left( \frac{E(X)}{n}\right)^2
\right)^{\frac{3}{2}}}\right)t^3\\&+\frac{1}{4!}\left( \frac{\sum\limits_{k=0}^{s}\left( k-\frac{E(X)}{n }\right)^4p^k q^{s-k}}{n\left( \sum\limits_{k=0}^{s}k^2p^kq^{s-k} -\left( \frac{E(X)}{n}\right)^2 \right)^2}\right)t^4e^{\psi(n)}.
\end{align}\]
Since \(\psi(n)\rightarrow0\) as \(n\rightarrow\infty\), it follows that \(\lim\limits_{n\rightarrow\infty}\Psi(n)=0\) for every fixed value of \(t\). Thus, \[\lim\limits_{n\rightarrow \infty} M_Z(t)=e^{t^2/2}, \text{ where } Z\rightsquigarrow N(0,1),\]
for all real values of \(t\). ◻
We can conclude that, as \(n\rightarrow\infty\), the random variable \(Z=\frac{X-E(X)}{\sigma_s}\) has the standard normal as its limiting distribution; or equivalently, that the bi\(^s\)nomial random variable \(X\) has, for large \(n\), an approximate normal distribution with mean \(\mu_s=np\frac{(s+1)p^{s}-1}{(p-q)}\) and variance \(\sigma_s^2=npq\frac{ 1-(s+1)^2(pq)^s}{(p-q)^2} .\)
\(^1\)USTHB, Faculty of Mathematics, RECITS Laboratory,
Po. Box 32, El Alia 16111, Bab Ezzouar, Algiers, Algeria
\(^2\)USTHB, Faculty of Mathematics, RECITS Laboratory,
Po. Box 32, El Alia 16111, Bab Ezzouar, Algiers, Algeria
E-mail addresses: hbelbachir@usthb.dz, hzeggada@usthb.dz