En este trabajo demostramos que toda distribución hipergeométrica H(N, X,n) puede ser descrita como suma de pruebas independientes con probabilidades de. View distribucion from CFM at Universidad Nacional de Colombia. DISTRIBUCIN HIPERGEOMTRICA. Notacin: Formula: Luego. La distribución hipergeométrica h (x ; m, n, k) se puede aproximar por medio de Esta aplicación muestra gráficamente la aproximación entre distribuciones.
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Nagar 1 and Danilo Bedoya Valencia 2. In this article, we have derived the probability density functions of the product and the quotient of two independent random variables having Gauss hypergeometric distribution. These densities have been expressed in terms of Appell’s first hypergeometric function F 1.
Appell’s first hypergeometric function, beta distribution, Gauss hypergeometric distribution, quotient, transformation. The beta distribution is very versatile and a variety of uncertainties can be usefully modeled by it.
Distrbucion of the finite range distributions encountered in practice can easily be transformed into the standard beta distribution. Several univariate and matrix variate generalizations of this distribution are given in Gordy 1.
A natural univariate generalization of the beta distribution is the Gauss hypergeometric distribution defined by Armero and Bayarri The above distribution was suggested by Armero and Bayarri 6. A brief introduction of this distribution is given in the encyclopedic work of Johnson, Kotz and Balakrishnan 3, p. In the context of Bayesian analysis of unreported Poisson hipergeomefrica data, while deriving the marginal posterior distribution of the reporting probablity p, Fader and Hardie 8.
The Gauss hypergeometric distribution has also been used by Dauxois 9.
Sarabia and Castillo This distribution was defined and used by Libby and Novic The beta distribution sometimes does not provide suficient flexibility for a prior for probability of success in a binomial distribution. Among various properties, the distribution defined by the density 5 can more flexibly account for heavy tails or skewness, and it reduces to the ordinary beta type 1 distribution for certain parameter choices.
The distribuion posterior distribution in this case is a four-parameter type of beta. Chen and Novic Several properties and special cases of this distribution are given in Johnson, Kotz and Balakrishnan 3, p. For further results and properties, the reader is referred to Aryal and Nadarajah In this article, we derive distributions of the product and the ratio of two independent random variables when at least one of them is Gauss hypergeometric.
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In this section, we give definitions and results that are used in subsequent sections. The integral representation of the Gauss hypergeometric function is given as. The Gauss hypergeometric function 2 F 1 a, b; c; z satisfies Euler’s relation.
For properties and further results the reader is referred to Luke The Appell’s first hypergeometric function F 1 is defined by.
It is straightforward to show that. For properties and further results of these function the reader is referred to Srivastava and Karlsson Now, using the definition of the Appell’s first hypergeometric function, we get the desired result.
The graph of the Gauss hypergeometric density for different values of the parameters is shown in the Figure 1. The k -th moment of the variable X having Gauss hypergeometric distribution, obtained in Armero and Bayarri 6.
Moreover, the cumulative distribution function CDF can be derived in terms of special functions as shown in the following theorem. Then, the CDF of X is given by. The CDF of X is evaluated as.
The following theorem suggests a generalized beta type 2 distribution, from the Gauss hypergeometric distribution. Let be distribhcion probability space. It is defined by.
This generalized entropy measure is given by. For details see Nadarajah and Zografos First, we give the following lemma useful in deriving these entropies.
To obtain 19we use 7 to write. In this section, we obtain distributional results for the product of two independent random variables involving Gauss hypergeometric distribution. Using the independence, the joint pdf of X 1 and X 2 is given by.
To find the marginal pdf of Zwe integrate 26 with respect to x 2 to get. Finally, applying 10we obtain the desired result. Since X 1 and X 2 are independent, their joint pdf is given by. Thus, we obtain the joint pdf of W and Z as. Finally, integrating w using 10 and substituting for K 2 in 28we obtain the desired result.
In this section we obtain distributional results for the quotient of two independent random variables involving Gauss hypergeometric distribution. In the following theorem, we consider the case where both the random variables are distrubucion as Gauss hypergeometric. The joint pdf of X 1 and X ditribucion is given by Now, using Lemma 2. Finally, using Lemma 2. Computationally convenient distributional assumptions for common-value auctions. Applied Probability and Statistics.
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Distributions of the product and the quotient of independent Kummer-beta variables. Prior assessments for predictions in queues. The special functions and their approximationsVol. Mathematics in Science and Engineering, Vol.
Fader and Bruce G. A note on modelling underreported Poisson counts.
Bayesian inference for linear growth birth and death processes. Bivariate distributions based on the generalized three-parameter beta distribution.
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Spanish pdf Article in xml format Article references How to cite this article Automatic translation Send this article by e-mail. Nagar 1 and Hiperggeometrica Bedoya Valencia 2 1 Ph. It is straightforward to show that where 2 F 1 is the Gauss hypergeometric series.
To obtain 19we use 7 to write and proceed similarly. All the contents of this journal, except where otherwise noted, is licensed under a Creative Hipergeomettrica Attribution License. How to cite this article.