EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 1, 2021, 192-203 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Note on the (Weighted) Bivariate Poisson Distribution R. Bidounga1,∗, P. C. Batsindila Nganga2, L. Niéré3, D. Mizère2 1 École Normale Supérieure, Université Marien Ngouabi, BP 69, Brazzaville, Congo 2 Faculté des Sciences et Techniques, Université Marien Ngouabi, BP 69, Brazzaville, Congo 3 Institut Supérieure de Gestion, Université Marien Ngouabi, BP 69, Brazzaville, Congo Abstract. In the recent statistical literature, the univariate Poisson distribution has been generalized by many authors, among them: the univariate weighted Poisson distribution [13], the generalized univariate Poisson distribution [7], the bivariate Poisson distribution according to Holgate [11], the bivariate Poisson distribution according to Lakshminarayana, Pandit and Srinivasa Rao [15], the bivariate Poisson distribution according to Berkhout and Plug [4], the bivariate weighted Poisson distribution according to Elion et al. [8] and the generalized bivariate Poisson distribution according to Famoye [9]. In this paper, We highlight the weighted bivariate Poisson distribution and show that it is the synthesis of all the bivariate Poisson distri- butions which, under certain conditions, converge in distribution towards the bivariate Poisson distribution according to Berkhout and Plug [4] which can be considered like the standard distribution in N2 as is the univariate Poisson distribution in N. 2020 Mathematics Subject Classifications: 62E10, 62E15, 62H10 Key Words and Phrases: Generalized Poisson distribution, convergence in distribution, conditional ditri- bution, bivariate generalized Poisson distribution, punctual duality 1. Introduction The bivariate Poisson distribution was discussed for the first time by Campbell [6] who consid- ered the limit of the distribution of a two-dimensional contingency table. Practically, at the same period Guldberg [10] obtains the bivariate distribution of independent Poisson distributions as the limit of the distribution of independent binomial distributions. The explicit form of the bivariate Poisson distribution is due a few years later to Aitken [1]. We had to wait Holgate [11] to obtain a bivariate Poisson variable from three independent univariate Poisson variables, i.e. with a non- diagonal variance-covariance matrix. A few years later, Kawamura [12] considered the structure of a bivariate Poisson distribution as the limit of a bivariate Bernoulli distribution and found the results of Holgate. We can refer to Morin [17] for a better edification. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i1.3895 Email addresses: rufbid@yahoo.fr (R. Bidounga), prevot.batsindila@gmail.com (P.C. Batsindila Nganga),leonard.niere@umng.cg (L. Niéré), domizere@gmail.com (D. Mizère). http://www.ejpam.com 192 c© 2021 EJPAM All rights reserved. R. Bidounga et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 192-203 193 Several authors have studied bivariate Poisson distributions, in particular Berkhout & Plug [4] and Lakshminarayana et al. [15]. Elion et al. [8] through the crossing of two weighted Poisson distributions revealed the bivariate weighted Poisson distribution. Batsindila Nganga et al. [2] showed that the bivariate Poisson distribution according to Holgate converges in distribution to the bivariate Poisson distribution according to Berkhout & Plug [4]. In this paper, we highlight the important role played by the bivariate Poisson distribution according to Berkhout & Plug [4] which allows to generate all the bivariate Poisson distributions. We show that the bivariate weighted Poisson distribution evidenced by Elion et al. [8] is a weighted bivari- ate Poisson distribution. We highlight the weighted bivariate Poisson distribution and show that it is the synthesis of all the bivariate Poisson distributions which, under certain conditions, converge in distribution towards the bivariate Poisson distribution according to Berkhout & Plug [4] which can be considered like the standard distribution in N2 as is the univariate Poisson distribution in N. The rest of this paper is organized as follows. In sections 2 and 3, we respectively recall the notion of univariate weighted Poisson distribution and the notion of generalized Poisson distribution. In section 4, we review the bivariate Poisson distributions according to Berkhout & Plug [4], according to Holgate [11], according to Lakshminarayana et al. [15] and the generalized bivariate Poisson distribution according to Famoye [9] then we show that these distributions converge in distribution towards the bivariate Poisson distribution according to Berkhout & Plug [4]. In section 5, we construct the weighted bivariate Poisson distribution and show that under certain conditions, this distribution is equal to the bivariate Poisson distribution according to Berkhout & Plug [4]. The section 6 presents the conclusion of this paper. 2. Univariate weighted Poisson distribution Suppose the realization y of the random variable Y of mass function p (y; δ) is recorded with a probability proportional to ω (y); the record y is the realization of a random variable Yω called weighted version of Y and which has the probability distribution: pω (y; δ) = P [ Yω = y ] = ω (y) Eδ [ω (Y)] p (y; δ) , y ∈ N := {0, 1, · · · } , δ ∈ R∗+ (1) called weighted distribution whereω (y) is called weight function, a positive function and Eδ [ω (Y)] = ∑ y∈N ω (y) p (y; δ) the constant of normalization which is the mean relative to the distribution of Y depending on δ such that 0 < Eδ [ω (Y)] < +∞. The function ω (y) = ω (y; φ) can depend on a parameter φ which represents the mechanism of saving data. Note that ω (y) = ω (y; δ, φ) can also depend on the canonical parameter δ. The data of a weight function makes it possible to generate a weighted probability distribution [13]. In this case, we can say that this distribution is generated by the weight function. In this paper, the distribution p (y; δ) will be called the basic distribution. When the basic distribu- tion is equal to the univariate Poisson distribution of parameter δ, the Expression (1) is called the univariate weighted Poisson distribution. The univariate weighted Poisson distribution has the following characteristics [3]: Eδ ( Yω) = δ ( 1 + d dδ lnEδ [ω (Y)] ) R. Bidounga et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 192-203 194 var ( Yω) = Eδ (Y) + δ2 d2 dδ2 lnEδ [ω (Y)] . Example 1. The univariate COM-Poisson distribution [5] with a probability mass function: P (Y = y|λ, ν) = λy (y!)ν 1 Z (λ, ν) , y = 0.1, . . . ; λ > 0, ν ≥ 0, is a univariate weighted Poisson distribution of weight function ω (y, ν) = (y!)ν−1 and constant of normalization: E [ω (Y, ν)] = e−λZ (λ, ν), with Z (λ, ν) = ∑+∞ n=0 λ n/ (n!)ν. 3. Generalized Poisson distribution The generalized Poisson distribution [7] of a random variable Y has the mass function: P (Y = y; δ, α) =  δy y! (1 + αy)y−1 e−δ(1+αy), y ∈ N 0 for y > m if α < 0, (2) with max ( −δ−1,−m−1 ) < α < δ−1, where m (≥ 4) is the largest positive integer such as 1+αm > 0, when α < 0. This distribution has the following characteristics [7]: Eδ (Y) = δ (1 − αδ)−1 var (Y) = δ (1 − αδ)−3 Eδ ( e−Y ) = eδ(s−1), with ln (s) − αδ (s − 1) + 1 = 0. 4. Bivariate Poisson distributions 4.1. Bivariate Poisson distribution according to Berkhout and Plug [4] Let Yi (i = 1, 2) a random variable which follows the univariate Poisson distribution with pa- rameter δi (i = 1, 2). The vector (Y1,Y2) follows the bivariate Poisson distribution according to Berkhout and Plug [4] if its mass function denoted fBP is equal to fBP (y1, y2; δ1, δ2) = δy1 1 y1! e−δ1  δy2 2 y2! e−δ2  , y1 ∈ N, y2 ∈ N, δ1 ∈ R ∗ +, δ2 ∈ R ∗ +, (3) under the conditions ln δ1 = x′β1 (4) and ln δ2 = x′β2 + ηy1, (5) R. Bidounga et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 192-203 195 where β1, β2 and η are parameters and x′ = ( x1, x2, . . . , xp ) the vector of deterministic variables or factors. The Expression (4) results in P (Y1 = y1; δ1) = ( δ y1 1 /y1! ) e−δ1 is a marginal distribution of Y1 and the Expression (5) means that P (Y2 = y2; δ2) = P (Y2 = y2/Y1 = y1) = ( δ y2 2 /y2! ) e−δ2 is a conditional probability. Thus, we have fBP (y1, y2; δ1, δ2) = P (Y1 = y1; δ1)P (Y2 = y2/Y1 = y1). When η = 0, then the variables Y1 and Y2 are independent. The generalized linear model of Expression (4) has for response variable Y1 and the model of Expression (5) has for response variable Y2. The resolution of these models makes it possible to highlight, not only the independence between the variables Y1 and Y2 but also the effect of the factor x′ on these same variables. The bivariate Poisson distribution according to [4] has the following characteristics [3]: Eδ1 (Y1) = var (Y1) = δ1 (6) Eδ2 (Y2) = ex′β2+c2+δ1(eη−1), (7) where c2 is the intercept of the model (5). var (Y2) = Eδ2 [Y2] + [ Eδ2 (Y2) ]2 ( eδ1(eη−1) − 1 ) (8) cov (Y1,Y2) = δ1Eδ2 [Y2] ( eη − 1 ) . (9) The expression (8) shows that the variable Y2 is overdispersed. The Expression (9) confirms the fact that the variables Y1 and Y2 are independent if and only if η = 0. And the covariance is negative, zero or positive depending on whether η is negative, zero or positive. 4.2. Bivariate Poisson distribution according to Holgate [11] Let be three univariate random variables V1, V2 and U independent of Poisson with respective parameters λ1, λ2 and λ3. With these three variables, we construct two new dependent variables Y1 and Y2 such as: Y j = V j + U, where j = 1, 2. (10) Then the joint distribution of the couple (Y1,Y2) is written: P (Y1 = y1,Y2 = y2) = e−λ1−λ2−λ3 min(y1,y2)∑ `=0 λ`3 `! λ y1−` 1 (y1 − `)! λ y2−` 2 (y2 − `)! ; y1, y2 = 0, 1, 2, . . . (11) By setting δ1 = λ1 + λ3 and δ2 = λ2 + λ3, we have the following result [2]: P (Y1 = y1,Y2 = y2) = δy1 1 y1! e−δ1  δy2 2 y2! e−δ2  × b (y1, y2; δ1, δ2, λ3) (12) with b (y1, y2; δ1, δ2, λ3) = eλ3 ( 1 − λ3 δ1 )y1 ( 1 − λ3 δ2 )y2 min(y1,y2)∑ `=0 (−y1)[`] (−y2)[`] z` `! (13) R. Bidounga et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 192-203 196 and z = λ3/ (δ1 − λ3) (δ2 − λ3) , (−y1)[`] = (−1)` y1!/ (y1 − `)!. We denote the distribution given by the Expression (11) by fH (y1, y2, λ1, λ2, λ3). The pair of variables (Y1,Y2) has the following characteristics [11]: Eδi (Yi) = var (Yi) = δi , (i = 1, 2) (14) cov (Y1,Y2) = λ3. (15) The marginal variable Yi (i = 1, 2) is a univariate Poisson variable with parameter δi (i = 1, 2). The variables Y1 and Y2 are dependent because their covariance is strictly positive. By taking δ y1 1 y1! e−δ1 = P [ Y1 = y1 ] , as the marginal distribution of Y1 and δ y2 2 y2! e−δ2 = P [ Y2 = y2/Y1 = y1 ] , as the conditional distribution of Y2 when we consider Y1 = y1, under the constraints (4) and (5), we find: P [ Y1 = y1,Y2 = y2 ] = P [ Y1 = y1 ] P [ Y2 = y2/Y1 = y1 ] , (16) P [ Y1 = y1,Y2 = y2 ] = P [ Y1 = y1 ] P [ Y2 = y2/Y1 = y1 ] = fBP (y1, y2; δ1, δ2) and fH (y1, y2; δ1, δ2, λ3) = fBP (y1, y2; δ1, δ2) × b (y1, y2; δ1, δ2, λ3) , (17) which are the results found by Batsindila Nganga et al. [2]. By setting λ3 = 1/n with n ∈ N∗, Batsindila Nganga et al. [2] constructed the family of bivariate Poisson distributions according to Holgate { fH,n/n ∈ N∗ } , with fH,n (y1, y2; δ1, δ2) = fH (y1, y2; δ1, δ2, 1/n). By making n tend to infinity, we have the following results [2]: lim n−→+∞ b (y1, y2; δ1, δ2, 1/n) = 1 (18) and lim n−→+∞ fH,n (y1, y2; δ1, δ2) = fBP (y1, y2; δ1, δ2) . (19) The bivariate Poisson distribution according to Holgate [11] converges in distribution to the bi- variate Poisson distribution according to Berkhout & Plug [4]. R. Bidounga et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 192-203 197 4.3. Bivariate Poisson distribution according to Lakshminarayana et al. [15] Lakshminarayana et al. [15] defined the bivariate Poisson distribution, which is the joint distribu- tion of the pair of random variables (Y1,Y2), as the product of Poisson marginal distributions with a multiplicative factor. The probability mass function of this bivariate Poisson distribution that we denote by fLPS is defined by: fLPS (y1, y2; δ1, δ2, λ) = δy1 1 y1! e−δ1  δy2 2 y2! e−δ2  [1 + λ ( e−y1 − e−dδ1 ) ( e−y2 − e−dδ2 )] , (20) with y1, y2 ∈ N, (δ1, δ2) ∈ ( R∗+ )2, λ ∈ R∗+ et d = 1 − e−1. This distribution has the characteristics (Lakshminarayana et al., 1999): Eδi (Yi) = δi, (i = 1, 2) cov (Y1,Y2) = δ1δ2d2e−c(δ1+δ2). The marginal variables are Poisson with parameters δi (i = 1, 2) et e−dδi = Eδi ( eYi ) (i = 1, 2). We have the following result. Proposition 1. Taking into account Expressions (3), (4) and (5), we have the following expression. fLPS (y1, y2; δ1, δ2, λ) = fBP (y1, y2; δ1, δ2) × ψ (y1, y2; δ1, δ2, λ) , (21) with ψ (y1, y2; δ1, δ2, λ) = 1 + λ ( e−y1 − e−dδ1 ) ( e−y2 − e−dδ2 ) . Proof. The proof is obvious. Corollary 1. By setting λ = λn, n ∈ N, such that limn−→+∞ λn = 0, we build a family of the bivari- ate Poisson distributions according to Lakshminarayana et al. [15], { fLPS ,n (y1, y2; δ1, δ2) /n ∈ N∗ } such that fLPS ,n (y1, y2; δ1, δ2) = fLPS (y1, y2; δ1, δ2, λn). We have limn−→+∞ ψ (y1, y2; δ1, δ2, λn) = 1 and therefore lim n−→+∞ fLPS ,n (y1, y2; δ1, δ2) = fBP (y1, y2; δ1, δ2) . (22) The bivariate Poisson distribution according to Lakshminarayana et al. [15] converges in distri- bution to the bivariate Poisson distribution according to Berkhout and Plug [4]. We can therefore notice, through Expression (21), that the bivariate Poisson distribution according to Lakshminarayana et al.[15] is the product of the bivariate Poisson distribution according to Berkhout & Plug with a multiplicative factor. The Expression (22) shows that the bivariate Poisson distribution according to Berkhout & Plug is a limit case of the bivariate Poisson distribution according to Lakshminarayana et al.[15] R. Bidounga et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 192-203 198 4.4. Bivariate generalized Poisson distribution Famoye [9] combines the generalized Poisson distribution of Consul & Jain [7] and the bivariate Poison distribution of Lakshminarayana et al. [15] to construct the distribution whose probability mass function is P (Y1 = y1,Y2 = y2) = 2∏ i=1 δyi i yi! (1 + αiyi)yi−1 e−δi(1+αiyi)  [1 + λ ( e−y1 − c1 ) ( e−y2 − c2 )] , (23) with ci = E ( e−Yi ) , yi ∈ N, δi ∈ R ∗ +, αi ∈ R, (i = 1, 2). We will denote the distribution given in Expression (23) by fF (y1, y2, δ1, δ2, α1, α2, λ). This dis- tribution has the following characteristics [9]: Eδi (Yi) = δi (1 − αiδi)−1 , i = 1, 2 var (Yi) = δi (1 − αiδi)−3 , i = 1, 2 cov (Y1,Y2) = λ (c11 − c1δ1) (c22 − c2δ2) , with cii = Eδi ( Yie−Yi ) = δi (1 − αiθisi)−1 eδi(1+αi)(si−1)−1 where ln (si)−αiθi (si − 1)+1 = 0 (i = 1, 2) . We have the following result. Proposition 2. Under the conditions (4) and (5), the Expression (23) becomes fF (y1, y2, δ1, δ2, α1, α2, λ) = fBP (y1, y2, δ1, δ2)ψF (y1, y2, δ1, δ2, α1, α2, λ) , (24) where ψF (y1, y2, δ1, δ2, α1, α2, λ) =  2∏ i=1 (1 + αiyi)yi−1 e−αiδiyi  [1 + λ ( e−y1 − c1 ) ( e−y2 − c2 )] . (25) Proof. Note that Expression (23) can still be written P (Y1 = y1,Y2 = y2) =  2∏ i=1 δ yi i yi! e−δi   2∏ i=1 (1 + αiyi)yi−1 e−δiαiyi  [1 + λ ( e−y1 − c1 ) ( e−y2 − c2 )] = δy1 1 y1! e−δ1  δy2 2 y2! e−δ2   2∏ i=1 (1 + αiyi)yi−1 e−δiαiyi  [1 + λ ( e−y1 − c1 ) ( e−y2 − c2 )] . Taking into account Expressions (3), (4) and (5), we get P (Y1 = y1,Y2 = y2) = fBP (y1, y2; δ1, δ2)  2∏ i=1 (1 + αiyi)yi−1 e−δiαiyi  [1 + λ ( e−y1 − c1 ) ( e−y2 − c2 )] . By setting ψF (y1, y2, δ1, δ2, α1, α2, λ) =  2∏ i=1 (1 + αiyi)yi−1 e−αiδiyi  [1 + λ ( e−y1 − c1 ) ( e−y2 − c2 )] , (26) R. Bidounga et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 192-203 199 it follows that fF (y1, y2, δ1, δ2, α1, α2, λ) = fBP (y1, y2, δ1, δ2)ψF (y1, y2, δ1, δ2, α1, α2, λ) . The proof is finished. Corollary 2. In Expression (26), let αi = αin, n ∈ N with limn−→+∞ αin = 0 (i = 1, 2) and λ = λn, n ∈ N with limn−→+∞ λn = 0. We can then build a family of Famoye distributions { fF,n/n ∈ N } such that fF,n (y1, y2, δ1, δ2, α1, α2, ) = fF (y1, y2, δ1, δ2, α1n, α2n, λn). Like limn−→+∞ ψF (y1, y2, δ1, δ2, α1n, α2n, λn) = 1, then lim n−→+∞ fF,n (y1, y2, δ1, δ2, α1, α2, ) = fBP (y1, y2, δ1, δ2) , the distribution of Famoye [9] converges in distribution towards the bivariate Poisson distribution according to Berkhout and Plug. Expression (24) confirms that the distribution evidenced by Famoye [9] is a bivariate Poisson distribution. 5. Weighted bivariate Poisson distribution Definition 1. Consider fBP (y1, y2; δ1, δ2) the basic distribution of the pair of random variables (Y1,Y2). We call the weighted bivariate Poisson distribution, the probability mass function defined by: fω (y1, y2; δ1, δ2, λ) = ω (y1, y2; δ1, δ2, λ) Eδ1,δ2 [ω (Y1,Y2; δ1, δ2, λ)] × fBP (y1, y2; δ1, δ2) , (27) where ω (y1, y2; δ1, δ2, λ) is called the weight function, a positive function, and Eδ1,δ2 [ω (Y1,Y2; δ1, δ2, λ)] = ∑ y1 ∑ y2 ω (y1, y2; δ1, δ2, λ) fBP (y1, y2; δ1, δ2) the constant of normalization such that 0 < Eδ1,δ2 [ω (Y1,Y2; δ1, δ2, λ)] < +∞. Let ψ (y1, y2; δ1, δ2, λ) = ω (y1, y2; δ1, δ2, λ) Eδ1,δ2 [ω (Y1,Y2; δ1, δ2, λ)] , (28) the normalized weight function ([16], [14]). The expression (28) results in ω (y1, y2; δ1, δ2, λ) = ψ (y1, y2; δ1, δ2) × Eδ1,δ2 [ω (Y1,Y2; δ1, δ2, λ)] . (29) From the Expression (29), we can deduce that the constant of normalization Eδ1,δ2 [ω (Y1,Y2; δ1, δ2, λ)] makes it possible to calculate the weight functions and consequently it also generates the weighted bivariate Poisson distribution. R. Bidounga et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 192-203 200 Example 2. suppose that ω (y1, y2; δ1, δ2, λ) = ω1 (y1)ω2 (y2) and Eδ1δ2 [ω (Y1,Y2; δ1, δ2, λ)] = Eδ1 [ω1 (Y1)]Eδ2 [ω2 (Y2)] . This last expression does not mean that the random variables Y1 and Y2 are independent. The mass function fω (y1, y2; δ1, δ2) given in Expression (27) is equal to: fω (y1, y2; δ1, δ2) = ω1 (y1) Eδ1 [ω1 (Y1)] ω2 (y2) Eδ2 [ω2 (Y2)] × fBP (y1, y2; δ1, δ2) . (30) The Expression (30) is the crossing between two univariate weighted Poisson distributions. It is called the bivariate weighted Poisson distribution [8]. Expression (30) shows that the bivariate weighted Poisson distribution is a weighted bivariate Poisson distribution. Its characteristics are ([3]): Eδ2 [ Yω2 2 ] = ex′β2+c2+δ1(eη−1)Eeηδ1 [ω1 (Y1)] Eδ1 [ω1 (Y1)] var ( Yω2 2 ) = Eδ2 [ Yω2 2 ] + [ Eδ2 ( Yω2 2 )]2 eδ1(eη−1)Eδ1 [ω1 (Y1)]Eδ1e2η [ω1 (Y1)]( Eδ1eη [ω1 (Y1)] )2 − 1  cov ( Yω1 1 ,Yω2 2 ) = Eδ2 [ Yω2 2 ] ( δ1eη + d dη ( lnEδ1eη [ω1 (Y1)] ) − Eδ1 [ Yω1 1 ]) . Proposition 3. If the univariate random variables Y1 and Y2 are punctually dual, then the bi- variate weighted Poisson distribution given by Expression (30) is equal to the bivariate Poisson distribution fBP (y1, y2; δ1, δ2). Proof. If Y1 and Y2 are punctually dual [13], then ω1 (y1)ω2 (y2) = 1, ∀ (y1, y2) ∈ N2. So Eδ1 [ω1 (Y1)]Eδ2 [ω2 (Y2)] = 1, therefore fω (y1, y2; δ1, δ2) = fBP (y1, y2; δ1, δ2). Example 3. In Expression (13), let ψ (y1, y2; δ1, δ2) = b (y1, y2; δ1, δ2, λ3) = eλ3 ( 1 − λ3 δ1 )y1 ( 1 − λ3 δ2 )y2 min(y1,y2)∑ `=0 (−y1)[`] (−y2)[`] z` `! . From the Expression (28), if we take Eδ1,δ2 [ω (Y1,Y2; δ1, δ2, λ)] = e−λ3 , as the constant of normalization, then the weight function is equal to (Cf. Expression (29)): ω (y1, y2; µ1, µ2, λ) = ( 1 − λ3 δ1 )y1 ( 1 − λ3 δ2 )y2 min(y1,y2)∑ `=0 (−y1)[`] (−y2)[`] z` `! . We deduce, from Definition 1, that the bivariate Poisson law according to Holgate [11] is a weighted bivariate Poisson distribution. R. Bidounga et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 192-203 201 Example 4. From Expression (21), we have ψ (y1, y2; µ1, µ2, λ) = 1 + λ ( e−y1 − e−dµ1 ) ( e−y2 − e−dµ2 ) , with d = 1 − e−1. If we take Eδ1,δ2 [ω (Y1,Y2; δ1, δ2, λ)] = e−(δ1−δ2)λ, as the constant of normalization, the weight function is equal to ω (y1, y2; µ1, µ2, λ) = [ 1 + λ ( e−y1 − e−dµ1 ) ( e−y2 − e−dµ2 )] e(δ1−δ2)λ. We deduce, from Definition 1, that the bivariate Poisson distribution according to Lakshminarayana and al. [15] is a weighted bivariate Poisson distribution. Example 5. Let Yi (i = 1, 2) be random variables of COM-Poison [5] with parameters (δi, νi) , (i = 1, 2). The COM-Poisson distribution is a weighted univariate Poisson distribution (Cf. Ex- pression (1)) with a weight function ωi (yi, δi) = (yi)1−νi , (i = 1, 2) , (31) and the constant of normalization Eδi [ω (Yi)] = e−δiZ (δi, νi) , (32) with Z (δi, νi) = ∑+∞ n=0 δ n i / (n!)νi . Taking into account the Expressions (31) and (32), the distribu- tion given by the Expression (30), called bivariate COM-Poisson distribution [5], is a weighted bivariate Poisson distribution of weight function ω (y1, y2, δ1, δ2, ν1, ν2) = (y1)1−ν1 (y2)1−ν2 and the constant of normalization Eδ1,δ2 [ ω (y1, y2, δ1, δ2, ν1, ν2) ] = Z (δ1, ν1) Z (δ2, ν2) e−δ1−δ2 . Example 6. From Expression (25), we have ψ (y1, y2, δ1, δ2, α1, α2, λ) =  2∏ i=1 (1 + αiyi)yi−1 e−αiδiyi  [1 + λ ( e−y1 − c1 ) ( e−y2 − c2 )] . If we take Eδ1,δ2 [ω (Y1,Y2; δ1, δ2, λ)] = 1, as the normalization constant and the weight function is equal to ω (y1, y2; δ1, δ2, λ) = ψ (y1, y2, δ1, δ2, α1, α2, λ) . The bivariate generalized Poisson distribution according to Famoye [9] is a weighted bivariate Poisson distribution, that is to say a bivariate Poisson distribution. REFERENCES 202 6. Conclusion We have reviewed the bivariate Poisson distributions and determined the functional relationships that exist between them. We have highlighted the important role played by the bivariate Poisson distribution according to Berkhout and Plug [4] which allows to generate all the bivariate Pois- son distributions. The bivariate weighted Poisson distribution evidenced by Elion et al. [8] is a weighted bivariate Poisson distribution. 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