COM-Poisson Beta Distribution

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1 COM-oisson Beta Distribution V.Saavithri Department of Mathematics Nehru Memorial College Trich. S.Thilagarathinam Department of Mathematics Nehru Memorial College Trich. Department of Mathematics Nehru Memorial College Trich. Abstract In this paper, we introduce COM-oisson Beta distribution. The new COM-oisson beta distribution densit function can be expressed as a mixture of COM-oisson and beta distribution densit function. Its properties are also studied. Ke Words: oisson distribution, Beta distribution, COM-oisson Distribution, COM-oisson Beta distribution. Introduction The Conwa-Maxwell oisson distribution discussed b Shmulli et.al.(25). This distribution as an extension of the oisson distribution is a popular model for analsing counting data. The COM-oisson distribution with parameter λ > and ν, sa COM (λ, ν). The case ν corresponding to the oisson distribution. The values of ν > correspond to under-dispersion, whereas the values of ν < represent over-dispersion with respect to the oisson distribution. When ν tends to infinitel, the Com-oisson distribution approaches to the Bernoulli distribution with parameter ( + λ). For ν and λ <, the COM-oisson distribution reduces to the geometric distribution with parameter λ. The beta distribution of the first kind, usuall written in terms of the incomplete beta function, can be used to model the distribution of measurements whose values all lie between zero and one. Gurand (958) made the further ISSN: age 2

2 assumption that the parameter p varies from cluster to cluster. This variation can be represented b a beta distribution. The distribution of the number of cluster Gurand assumed also that the number of egg masser per plot is oisson. The beta binomial model for the distribution is used ver widel. Muench (963, 938) was interested in applications to medical trials. Skellam (948) applied the model to the association of chromosomes and to traffic clusters. Guenther (97) showed that the average cost per lot for the Hald linear cost model with a beta prior has a beta-binomial distribution. The beta-binomial distribution has itself been used as a prior distribution b Steck and Zimmer (968). In this paper, we introduce COM-oisson Beta distribution. The new COMoisson beta distribution densit function can be expressed as a mixture of COM-oisson and beta distribution densit function. Its properties are also studied. In section 2,3, COM-oisson and Beta Distributions are defined. In section 4, COM-oisson Beta Distribution is defined.its properties are derived in section 5. 2 COM-oisson Distribution The probabilit densit function of COM-oisson distribution is (X x) where λj ν j (j!) λx, (x!)ν x,, 2,... for λ > and ν. The probabilit generating function of COM-oisson distribution is X (s). 3 Beta Distribution The Beta distribution is given b f (x; α, β) xα ( x)β where the parameters α and β are positive real quantities and the variable x satisfies x. The quantit is the Beta function defined in terms ISSN: age 3

3 of the more common Gamma function as Γ(α)Γ(β) Γ(α + β) For α β the Beta distribution simpl becomes a uniform distribution between zero and one. 4 Com-oisson Beta Distribution The COM-oisson beta distribution arises in a model formed b supposing that objects (which are to be countable) occur in clusters. Suppose there are Y independent random variables of the form X, and N denotes the sum of these random variables, namel N X + X2 +...XY Then, the COM-oisson beta distribution model is derived b the following assumptions (i) X represents the number of objects with in a cluster where X follows COMoisson distribution with parameters λ, ν. (ii) Y represents the number of clusters, where Y follows beta distribution with parameters α, β. This random variable N, formed b compounding in this fashion gives rise to the Com-poisson beta distribution and its probabilit generating function can be derived easil. The probabilit generating function of X is known to be GX (s) () The probabilit generating function of Y is Z GY (s) s α ( )β Since Xi s are iid and independent of Y, the robabilit generating function of ISSN: age 4

4 the random variable N can be derived as follows GN (s) E(sN ) E sx +X X X E sx +X X Y (Y ) X [E(s)] (Y ) X [GX (s)] (Y ) GY (GX (s)) Z (GX (s)) α ( )β Z GN (s) α ( )β (2) Now, since the probabilit generating function of N in (2) can be expressed as Z α Z β ( ) (λs)j α ( )β () j upon collecting the coefficient of sm in the above series, we find an explicit expression for the robabilit mass function of N as. Z (N m) (λ)m α ( )β () m! Z (N x) α ( )β (λ)x () x! This is the probabilit mass function of the Com-isson beta distribution and denote it b N C B(λ, ν, α, β) ISSN: age 5

5 5 roperties The mean and variance can be calculated from the first and second derivatives of the probabilit generating function b setting s GN (s) Z j j(λs)j λ () α ( )β Mean GN () Z j j(λ)j () α ( )β GN (s) Z j j(j )(λs)j 2 (λ)2 () α ( )β GN () Z j j(j )(λ)j () α ( )β V ar(n ) GN () + GN () [GN ()]2 Z V ar(n ) j j(j )(λ)j () α ( )β Z + 2 j(λ)j Z ν j (j!) α ( )β α ( )β j () () j(λ)j From these,we find the ratio between mean and variance to be R V ar(n ) M ean j (Z(λ,ν)) R ISSN: j(j )(λ)j j j(λ)j (Z(λ,ν)) α ( )β B(α,β) Z α ( )β B(α,β) j j(λ)j () α ( )β + 5 age 6

6 References [] Gurland J (958):A generalized class of contagious distribution Biometrics, 4, [2] Guether W. C (97): The average sample number for truncated double sample attribute plans, technometrics, 3, [3] Muuench H (938): Discrete frequenc distributions arising from mixtures of several single probabilitnvalues, journal of American Statistical Association, 33, [4] Shmeli G, Minka T., Kadane J.B, Borle S and Boatwright (25): A useful distribution for fitting discrete data:revival of the COM-oisson ditribution, J.R.Stat.Soc.Ser.C(Appl. Stat), 54, [5] Skellam J.G (948), A probabilit distribution derived from the binomial distribution b regarding the probabilit of success as variable berween the sets of trials, Journal of the Roal Statistical Societ, series B,, [6] Steck G.., and Zimmer W.J (968) The relationship between Neman and Baes confidence intervals for the hpergeometric parameter, Technometrics,, [7] riadharshini J., and Saavithri,V., COM-oisson ola-aeppli rocess, aper communicated to the journal of Statistics and probabilit letters. ISSN: age 7

Compound COM-Poisson Distribution with Binomial Compounding Distribution

Compound COM-Poisson Distribution with Binomial Compounding Distribution Compound COM-oisson Distribution with Binomial Compounding Distribution V.Saavithri Department of Mathematics Nehru Memorial College Trichy. saavithriramani@gmail.com J.riyadharshini Department of Mathematics