Exact Distribution of Random Weighted Convolution of Some Beta Distributions Through an Integral Transform

We give the exact distribution of the average of n independent beta random variables weighted by the selected cuts of (0,1) by the order statistics of a random sample of size 1 n from the uniform distribution (0,1) U , for each n . A new integral transformation that is similar to generalized Stieltjes transform is given with various properties. Integral representation of the Gauss-hypergeometric function in some parts is employed to achieve the exact distribution. Also the result of Soltani and Roozegar [On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach. Statist Probab Lett. 2012;82(5):1012-1020] with the new transform is achieved. Finally, several new examples of this family of distributions are investigated.


Introduction
Van Assche (1987) defined a random variable T uniformly distributed between two independent random variables.He employed the Stieltjes transform and derived that: (i) for 1 X and 2 X on 1,1] [ , T is uniform on 1,1] [ if and only if 1 X and 2 X have an arcsine distribution; and (ii) T have the same distribution as 1 X and 2 X if and only if they are degenerated or have a Cauchy distribution.After that, Johnson and Kotz (1990) showed that the random variable T is a weighted average of 1 X and 2 X with random weights W and W  1 ; , and W is uniform on [0,1] independent of 1 X and 2 X .They also extended the randomly weighted averages to more than two random variables with Dirichlet (1,1,...,1) random weights.In fact they neither proved nor disapproved Van Assche's results.Moreover, Johnson and Kotz (1990) discuss on calculating distribution of randomly weighted averages of two independent random variables ( 2 = n ) not for general n .Also, they cannot find the distribution of randomly weighted averages for general n ( 2 > n ) and there is not any example there for general n .Soltani and Homei (2009) followed Johnson and Kotz (1990) and investigated Van Assche's results.Soltani and Roozegar (2012) employ certain techniques in divided differences to relate the generalized Stieltjes transform of the distribution of a randomly weighted average of independent random variables m X X ,..., Recently, many works have been done on randomly weighted averages.Di Salvo (2008) applied the theory on multiple hypergeometric functions and find the distribution of a weighted convolution of gamma variables, Roozegar and Soltani (2014) determine certain classes of power semicircle distributions, that are randomly weighted average distributions.The brief and direct proof of the main result in Roozegar and Soltani (2014) is given in Roozegar and Soltani (2013).A certain version of randomly weighted averages of two independent and continuous random variables with beta random weights is provided in Roozegar (2014).A relation between the Cauchy-Stieltjes transforms of the distribution functions of randomly weighted averages with Dirichlet random proportions and n independent random variables 1 ,..., n XX is found in Roozegar (2015).Roozegar and Soltani (2015) investigate the asymptotic behavior of randomly weighted averages.Demni (2016) provide the generalized Stieltjes transforms of some compactlysupported probability distributions with a lot of examples.In Demni (2016) we can find some resaons for importance of convolution of beta distributions.In other work, Roozegar (2017) find the limiting behavior of randomly weighted averages in the case of symmetric heavy-tailed random variables.

U
. Randomly weighted average (RWA) of independent and continuous random variables where the proportions weights.Suppose that among the above whole order statistics, we select , where 2 ≤ k ≤ n and  0 = 0 < 1 <  2 < K <   = .A general form to n S will be the RWA of k independent and continuous random variables  1 , … ,   , denoted by  : 1 ,…, −1 , which is given by 2) where the random weights j V are defined by   =  (  ) −  ( −1 ) ,  = 1,2, … , .
In the above we have put the conventions 0 The RWA on the form (1.2) was defined and studied by Soltani and Roozegar (2012).In fact the random vector V= (  1 ,  2 , … ,   ) has the Dirichlet distribution, Dir (  1 ,  2 , … ,   ), with the probability density function (pdf) and zero outside.
To state assertions, we introduce beta distributions, ̃(, ) over ] , [ b a by the pdf is beta function.
Let us denote the usual beta distribution over The ordinary Stieltjes transform (ST) and generalized Stieltjes transform (GST) of a distribution H are respectively defined by where £ is the set of complex numbers, supp H stands for the support of H and  is a constant.For more on the ST and GST, see Debnath The Gauss-hypergeometric function (1)   D F , is defined by the series , ! ) ( denotes the rising factorial.
Gauss-hypergeometric function (1)   D F has the Euler's integral representation of the form . ) (1

New transform and RWA distribution
First, we define a new univariate characteristic function called an additive Stieltjes transform (AST).

Definition 1. If X is a random variable with distribution H on a subset S of
where d is a positive real number.
The assumptions that d is positive and H has a support in S are needed for the one to one correspondence between AST and H in next theorem.
Although the AST and GST transforms have different domains of definition but analytically they have the following relationship which will be useful for the rest of the paper.Specially this issue is not important for next two family of randomly weighted averages.The distribution H is assumed to produce convergent both integrals as mentioned above for each â z C  so that the function . For more information and detailed overview of the properties of GST, see Karp  .
Proof.Applying Theorem 3.1 of Soltani and Roozegar (2012) and then (2.1) to conclude the result.

Families of RWA on some beta distributions
In this section, we provide some families of RWA  .
By Table 3 in Kubo et al. (2011), it follows that Another general family of RWA on beta distributions on ] , [ b a is presented in the following theorem.The following corollary is an immediate consequence of Theorem 4.

Corollary 1. For independent random variables
The following theorem of Roozegar and Soltani (2014) comes as a corollary to Corollary 1.

Corollary 2. For every integer n≥2, a power semicircle distribution with shape parameter
3)/2 ( =  n  and any positive range parameter is a RWA n S distribution.

Proof. Consider
, then use Corollary 3.1 to obtain the result.

Examples
Theorem 3 and Theorem 4 in Section 3 give general results on distribution of RWA distribution.

Conclusions
In this paper, We compute the exact distribution of the weighted average of n independent beta random variables where the weightes are the selected cuts of (0,1) by the order statistics of a random sample of size 1 n  from the uniform distribution.We provide A new integral transformation with some of its mathematical properties.Integral representation of the Gauss-hypergeometric function in some parts is employed to achieve the exact distribution.Finally we investigate several new examples of this family of distributions.

1 to 1 ;
the generalized Stieltjes transforms of the distribution functions m F F ,...,   ~ , m i 1,..., = .The random weights are assumed to be cuts of [0,1] by 1  m (selected) ordered statistics of independent and identically uniformly distributed random variables n U U ,..., 1 on [0,1]; m≤ .Soltani and Homei (2009) treated the case n m = using the Schwartz distribution theory.In this paper, we find the distribution of randomly weighted averages for general n and for Dirichlet new transform (similar to the Stieltjes transform).Finding examples is difficult, but we identified fairly large classes of randomly weighted average of common beta distributions with Dirichlet ) integral representation of the Gauss-hypergeometric function and the new transform.

.
The power semicircle distribution with parameters  and  on ( special case of beta distribution whenever  =  =  +

4 )
For more details on Gauss-hypergeometric function and its properties, seeAbramowitz and Stegun (2012) andAndrews et al. (1999).There are few examples of known distributions that are RWA distributions of the form (1.1), n S , and there are no examples, to the best of our knowledge, that are RWA distributions of the form (1Soltani (2013) used the investigations ofDemni (2009) andKubo et al. (2011) on the connection between ST and GST of two distributions and introduced new classes of power semicircle laws that are RWA n S distributions.In this paper we introduce new classes of RWA 1 paper is organized as follows.In Section 2 we introduce a new transform similar to GST for later analysis.We rewrite the main result ofSoltani and Roozegar (2012) based on this new transform as Theorem 2.2 in this section.Two new classes of RWA in Section 3. In Section 4, we present some examples of the new classes.