Recently, Alkarni and Oraby (2012) obtained general forms for some properties of the compound class of Poisson and lifetime (PL) distributions. In this paper, we obtain some general forms for joint density, cumulative distribution, and survival functions of the bivariate case of PL class. Its conditional distributions are also studied. In addition, the compound class of geometric and lifetime distributions as well as its mixed bivariate case are discussed. For this class some conditional probabilities useful for reliability, biological survey, and engineering are also studied. Our class contains several new mixed bivariate distributions in special cases

Bidram, Hamid

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J. Stat. Appl. Pro. 2, No. 1, 21-25 (2013) 21Journal of Statistics Applications & ProbabilityAn International Journalc⃝ 2013 NSPNatural Sciences Publishing Cor.A bivariate compound class of geometric–Poisson andlifetime distributionsHamid BidramDepartment of Statistics, University of Isfahan, Isfahan, IranReceived: 7 Sep. 2012, Revised: 8 Oct. 2012, Accepted: 2 Jan. 2013Published online: 1 Mar. 2013Abstract: Recently, Alkarni and Oraby (2012) obtained general forms for some properties of the compound class of Poisson andlifetime (PL) distributions. In this paper, we obtain some general forms for joint density, cumulative distribution, and survivalfunctions of the bivariate case of PL class. Its conditional distributions are also studied. In addition, the compound class of geometricand lifetime distributions as well as its mixed bivariate case are discussed. For this class some conditional probabilities useful forreliability, biological survey, and engineering are also studied. Our class contains several new mixed bivariate distributions in specialcases.Keywords: Compound distribution, Geometric distribution, Poisson distribution, Mixed bivariate distribution, Random sample size1 IntroductionLet X1,X2, ...,XN be N independent and identically distributed (iid) random variables from an absolutely continuousdistribution F , where N has a geometric or Poisson distribution and is independent of Xi’s and also let U = min{Xi}Ni=1. Inrecent years, several compound lifetime distributions have been introduced by U in the literature. Exponential-geometric(EG) distribution of Adamidis and Loukas (1998), exponentiated exponential-geometric (E2G) distribution of Louzada(2012), Weibull-geometric (WG) distribution of Barreto-Souza et al. (2011), exponential-Poisson (EP) distribution ofKus (2007), and Weibull-Poisson (WP) distribution of Hemmati et al. (2011) are the remarkable distributions in thisconnection.Recently, Alkarni and Oraby (2012) obtained some general forms for density, cumulative distribution, survival,and hazard rate functions of U , when N has a Poisson distribution. In this paper, we study its mixed bivariate randomvariable, i.e., (U,N). In addition, we obtain general forms for density, cumulative distribution, survival, and hazard ratefunctions of U , when N has a geometric distribution and also study its mixed bivariate case.2 General expressions for mixed bivariate distributions2.1 A geometric distribution for NIn this section, we consider a geometric distribution for N with probability mass functionP(N = n) = pqn−1, n = 1,2, ..., (0 < q = 1− p < 1),denoted by N ∼ Ge(p), and then we obtain a set of theorems, relations, and general forms for some properties of U andfor mixed bivariate random variables (U,N).Definition 2.1 Let X1,X2, ...,XN be iid random variables from an absolutely continuous distribution with cumulativedistribution function (cdf) F, survival function F, and probability density function (pdf) f , where N ∼ Ge(p) isindependent of Xi’s. Then, we say that U = min{Xi}Ni=1 has a F-geometric distribution and will be denoted byU ∼ FG(θ , p), where θ is the parameter vector of F.∗ Corresponding author e-mail: hamid−bidram@yahoo.com (h.bidram@sci.ui.ac.ir)c⃝ 2013 NSPNatural Sciences Publishing Cor.22 Hamid Bidram: A bivariate compound class of geometric...Theorem 2.1 Let U ∼ FG(θ , p). Then, the cdf, survival (reliability) function, pdf, and hazard rate function of U, for allx ∈ R+, are given byFU (x) =F(x)1−qF(x), (1)FU (x) =pF(x)1−qF(x), (2)fU (x) =p f (x){1−qF(x)}2(3)andhU (x) =h(x)1−qF(x), (4)respectively, where h(x) = f (x)F(x) is the hazard rate function of X ∼ F.Proof. Using a conditional argument on N, the proof is completed. (See also Marshall and Olkin, 1997).Remark 2.1 As we see from 4, 11−qF(x) is a decreasing function for all x ∈ R+. Thus, hU (x) is decreasing if h(x) isdecreasing. Indeed, it says a mixture of decreasing hazard rate functions is decreasing. (See Barlow and Proschan, 1975).One can show that the properties given in Theorem 2.1 agree with corresponding properties of EG, WG, and E2Gdistributions as special cases.Definition 2.2 Under conditions given in Def. 2.1, a random vector (U,N) with stochastic representation(U,N) d= (min{Xi}Ni=1,N) is said to have a mixed bivariate F-geometric distribution and will be denoted by(U,N)∼ BFG(θ , p).Theorem 2.2 Let (U,N)∼ BFG(θ , p). Then, the joint pdf of (U,N) for x ∈ R+ and n = 1,2, ..., is given byfU,N(x,n) = n f (x)F(x)n−1 pqn−1. (5)Proof. We write fU,N(x,n) = ∑∞n=1 fU |N(x).P(N = n) and then the proof is completed.Theorem 2.3 Let (U,N)∼ BFG(θ , p). Then, the conditional pdf of N|U = x for x ∈ R+ and n = 1,2, ..., is given byfN|U=x(n) = n[1−qF(x)]2[qF(x)]n−1. (6)Proof. It is clear that fN|U=x(n) =fU,N(x,n)fU (x). Replacing fU,N(x,n) and fU (x) by 5 and 3, respectively, completes the proof.Corollary 2.1 (N|U = x) d=W −1, where W ∼ nb(2,1−qF(x)). ”nb” denotes the negative binomial distribution.Corollary 2.2 The moment generating function (mgf), mean, and variance of N|U = x, for x ∈ R+, are given byMN|U=x(t) = et{1−qF(x)1−qF(x)et }2, t <− logqF(x), E(N|U = x) = 1+qF(x)1−qF(x) , and Var(N|U = x) =2qF(x){1−qF(x)}2 , respectively.Proof. Using Corollary 2.1 completes the proof.Theorem 2.4 Let (U,N)∼BFG(θ , p). Then, the joint cdf and joint survival function of (U,N), for x∈R+ and n= 1,2, ...,are given byFU,N(x,n) = FN(n)−FU (x)FN◦(n) (7)andFU,N(x,n) = FU (x)FN◦(n), (8)respectively, where FN(n) = 1−qn is the cdf of N ∼ Ge(p), FU (x) is given in (2), and FN◦(n) = 1−FN◦(n) = 1− [qF(x)]nis the cdf of N◦ ∼ Ge(1−qF(x)).c⃝ 2013 NSPNatural Sciences Publishing Cor.J. Stat. Appl. Pro. 2, No. 1, 21-25 (2013) / www.naturalspublishing.com/Journals.asp 23Proof.FU,N(x,n) = P(U ≤ x,N ≤ n) =n∑k=1∫ x0f(U,N)(t,k)dt =n∑k=1∫ x0k f (t)F(t)k−1 pqk−1dt=n∑k=1pqk−1{1−F(x)k}= FN(n)−FU (x)FN◦(n).A similar proof is applied to FU,N(x,n) and then the proof is completed. Here, the identity FU,N(x,n) = 1−FU (x)−FN(n)+FU,N(x,n) can also be applied.Corollary 2.3 N◦d= (N = k|U > x), for k = 1,2, ..., where N◦ ∼ Ge(1−qF(x)).Proof. The proof is straightforward and so is omitted.Some conditional probabilities are useful for reliability, survival analysis, biological survey, and engineering.Here, we present some conditional probabilities below:Theorem 2.5 Let (U,N)∼ BFG(θ , p). Then, for x ∈ R+ and n=1,2,...P(U > x,N > m|N > n) ={qm−nFU (x)F(x)m, m ≥ n > 0FU (x)F(x)n, 0 < m ≤ n.Proof. For m ≥ n > 0:P(U > x,N > m|N > n) = P(U > x,N > m,N > n)P(N > n)=P(U > x,N > m)P(N > n)=FU (x)FN◦(m)qn= qm−nFU (x)F(x)m.A similar proof is applied to the case 0 < m ≤ n.Theorem 2.6 Let (U,N)∼ BFG(θ , p). Then, for x ∈ R+ and n=1,2,...P(U > x,N > n|U > u) ={FU (x)FN◦ (n)FU (u), 0 < u ≤ x[qF(u)]n = P(U > u|N > n), 0 < x ≤ u.Proof. The proof is similar to the proof of Theorem 2.5 and so is omitted.2.1.1 A special caseLet X1,X2, ...,XN be N iid random variables from an exponential distribution with the cdfF(x) = 1− e−βx, x > 0, β > 0, where N ∼ Ge(p) is independent of Xi’s. According to given relations in the previoussection, we have the following results.1. According to Theorem 2.1, the cdf, survival function, and pdf of U =min{Xi}Ni=1 are given by FU (x) =1−e−βx1−qe−βx , x >0, FU (x) =pe−βx1−qe−βx , and fU (x) =pβe−βx{1−qe−βx}2 , respectively. Further, Eq. (4) implies that the hazard rate function of U isdecreasing, since h(x) of exponential distribution is constant and 11−qF(x) is always decreasing.Remark 2.2 As we see, the above results agree with the EG distribution of Adamidis and Loukas (1998).In the sequel, we consider the mixed bivariate exponential-geometric distribution and denote it by(U,N) ∼ BEG(β , p). It is clear that the EG distribution of Adamidis and Loukas (1998) is the marginal distribution ofBEG(β , p) distribution.2. According to Theorem 2.2, the joint pdf of BEG(β , p) distribution is given byf(U,N)(x,n) =nβpq(qe−βx)n, x > 0, n = 1,2, ... .c⃝ 2013 NSPNatural Sciences Publishing Cor.24 Hamid Bidram: A bivariate compound class of geometric...3. According to Corollary 2.1, the conditional pdf of N|U = x is given byfU |N(x) = n(1−qe−βx)2(qe−βx)n−1, x > 0, n = 1,2, ... .Indeed (N|U = x) d=W −1, where W ∼ nb(2,1−qe−βx).4. According to Theorem 2.4, the joint cdf and survival function of (U,N) are given byFU,N(x,n) =(1−qn)(1−qe−βx)− pe−βx(1− (qe−βx)n)1−qe−βx, x > 0, n = 1,2, ...andFU,N(x,n) =pqne−(n+1)βx1−qe−βx, x > 0, n = 1,2, ...,respectively.In addition, we can obtain the following conditional probabilities having applications in reliability and survivalanalysis.5.P(U > x,N > m|N > n) =pqm−ne−(m+1)βx1−qe−βx , m ≥ n > 0pe−(n+1)βx1−qe−βx , 0 < m ≤ n.6.P(U > x,N > n|U > u) ={e−(β−u)x(1−qe−βu)qne−nβx1−qe−βx , 0 < u ≤ x(qe−βu)n = P(U > u|N > n), 0 < x ≤ u.Remark 2.3 The above properties can be easily applied to (U,N) ∼ BWG(α ,β , p), where BWG denotes the mixedbivariate Weibull-geometric distribution. It is enough that e−βx is replaced by e(−βx)α, which is the cdf of a Weibulldistribution. In addition, if F(x) = (1− e−βx)α , i.e., the cdf of exponentiated exponential distribution, then we have amixed bivariate exponentiated exponential-geometric distribution, which one can check all above mentioned propertiesfor this new distributions.2.2 A Poisson distribution for NHere, we consider a zero truncated Poisson distribution with rate λ for N with pmfπN(n) =e−λ λ nn!(1− e−λ ), n = 1,2, ... . (9)We denote it by N ∼ Po(λ ). The properties of U = min{Xi}Ni=1 has already been studied by Alkarni and Oraby (2012). Inthis section, we study the properties of the mixed bivariate random variable (U,N).Definition 2.3 Under same conditions given in Def. 2.1, let N ∼ Po(λ ) is independent of Xi’s. Then, the random vector(U,N) is said to have a mixed bivariate F-Poisson distribution and will be denoted by (U,N) ∼ BFP(θ ,λ ), where θ isthe parameter vector of F.Theorem 2.7 Let (U,N)∼ BFP(θ ,λ ). Then, the joint pdf of (U,N) for x ∈ R+ and n = 1,2, ..., is given byfU,N(x,n) = n f (x)[F(x)]n−1e−λ λ nn!(1− e−λ ). (10)Proof. Writing fU,N(x,n) = ∑∞n=1 fU |N(x)P(N = n) completes the proof.Theorem 2.8 Let (U,N)∼ BFP(θ ,λ ). Then, the conditional pdf of N|U = x for x ∈ R+ and n = 1,2, ..., is given byfN|U=x(n) =[λF(x)]n−1e−λF(x)(n−1)!. (11)Corollary 2.4 (N|U = x) d=W1 +1, where W1 ∼ P(λF(x)).Corollary 2.5 The mgf, mean, and variance of N|U = x, for x ∈ R+, are given by MN|U=x(t) = et+λF(x)(et−1), E(N|U =x) = 1+λF(x), and Var(N|U = x) = λF(x), respectively.c⃝ 2013 NSPNatural Sciences Publishing Cor.J. Stat. Appl. Pro. 2, No. 1, 21-25 (2013) / www.naturalspublishing.com/Journals.asp 252.2.1 Two special casesLet X1,X2, ...,XN be N iid exponential random variables, where N ∼ Po(λ ). According to mentioned properties inprevious section, we can have a new mixed bivariate distribution, called mixed bivariate exponential-Poisson(BEP(β ,λ )) distribution. Further, by replacing a Weibull distribution with parameters α and β instead of exponentialdistribution, we have another new mixed bivariate distribution, called mixed bivariate Weibull-Poisson (BWP(α,β ,λ ))distribution. It is clear that the marginal continuous distributions of BEP(β ,λ ) and BWP(α,β ,λ ) are, respectively, theEP distribution of Kus (2007) and WP distribution of Hemmati et al. (2011).Remark 2.4 All properties given in previous sections can be extended to the mixed bivariate random variable (V,N),where V = max{Xi}Ni=1.3 Concluding remarksIn this paper, we studied the mixed bivariate random variable (U,N). Many applications can be considered when therandom sample size is itself a random variable. For example, suppose that N is the number attacks of a forward playerduring a soccer which has a discrete distribution. Let Xi’s be the used times for every attack having an absolutelycontinuous distribution F . If forward player attacks m times in a soccer and U = min{Xi}Ni=1, then(u1,n1),(u2,n2), ...,(um,nm) are m joint observations of this example. As an alternative example, suppose that theadvertising number of a big trading company in media has a discrete distribution and the amount of earned gain arisingthe advertising has an absolutely continuous distribution F . It is here interesting to consider V = max{Xi}Ni=1, then for mtimes attempt we have (v1,n1),(v2,n2), ...,(vm,nm) observations. As other examples, we can see such applications inwater resources and climate studies focus on the understanding and prediction of properties of hydroclimatic episodes.For more real examples and also for some applications of BFG models, see Kozubowski and Panorska (2008).AcknowledgementThe author would like to sincerely thank an anonymous referee for carefully reading the paper and giving usefulsuggestions.References[1] Alkarni, S., Oraby, A. A compound class of Poisson and lifetime distributions. Journal of Statistics Applications and Probability1, 45-51, (2012).[2] Adamidis, K., Loukas, S. A lifetime distribution with decreasing failure rate. Statistics and Probability Letters 39, 35-42, , (1998).[3] Barlow, E., Proschan, F. Statistical theory of reliability and life testing probability models. Holt, Rinehart and Winston Inc, NewYork, (1975).[4] Barreto-Souza, W., Lemos-Morais, A., Cordeiro, G.M. The Weibull-geometric distribution. Journal of Statistical Computation andSimulation 81, 645-657, (2011).[5] Hemmati, F., Khorram, E., Rezakhah, S. A new three-parameter ageing distribution. Journal of Statistical Planning and Inference141, 2266-2275, (2011).[6] Kozubowski, T.j., Panorska, A.K. A mixed bivariate distribution connected with geometric maxima of exponential variables.Communications in Statistics-Theory and Methods 37, 2903-2923, (2008).[7] Kus, C. A new lifetime distribution. Computational Statistics and Data Analysis 51, 4497-4509, (2007).[8] Louzada, F., Marchi, V.A.A, Roman, M., 2012. The exponentiated exponential-geometric distribution: a distribution withdecreasing, increasing and unimodal failure rate. Statistics. DOI:10.1080/02331888. 2012.667103.[9] Marshall, A.W., Olkin, I. A new method for adding parameters to a family of distributions with application to the exponential andWeibull families. Biometrica 84, 641-652, (1997).c⃝ 2013 NSPNatural Sciences Publishing Cor.

A bivariate compound class of geometric–Poisson and lifetime distributions

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