A GENERALIZED BETA FUNCTION AND ASSOCIATED PROBABILITY DENSITY

We introduce and establish some properties of a generalized form of the beta function. Corresponding generalized incomplete beta functions are also defined. Moreover, we define a new probability density function (pdf) involving this new generalized beta function. Some basic functions associated with the pdf, such as moment generating function, mean residue function, and hazard rate function are derived. Some special cases are mentioned. Some figures for pdf, hazard rate function, and mean residue life function are given. These figures reflect the role of shape and scale parameters.

Definition 1.1. Continue with the preceding assumptions on the parameters a, b, c, ω, u, and v. Then for Re(a + µ) and Re(b + µ) > 0, we define a generalized form of the beta function as where ω 2 R 1 (a, b; c; x) is the ω-Gauss hypergeometric function [12,13] whose series representation is given by, Γ (c + ωk) x k k! , |x| < 1 (1.3) and its integral representation in the form, where . (1.5) For ω = 1, (1.4) is the classical Gauss hypergeometric function, hence (1.2) becomes Further, if we take a = 0 then (1.2) reduces to the well-known beta function, that is, In Section 2, we establish a number of analytic properties, such as recurrence relations and the asymptotic expansions for our generalized beta function. We express this generalized beta function in terms of ω-hypergeometric functions [13] In Section 3, we define generalized incomplete beta functions associated with the function defined by (1.2). These incomplete forms of the generalized beta functions are used to study some statistical functions in later sections. Moreover, we introduce and study a new probability density function (pdf) involving the generalized beta function in Section 4. Finally, we derive some basic functions associated with this density function, namely, the kth moment, moment generating function, the hazard rate function, and the mean residue life function. Corresponding results for beta distribution are listed. Some figures are given for pdf, hazard rate function, and mean residue life function.
2. The generalized beta function. We begin this section by observing that (1.2) can be rewritten as We express this generalized beta function in terms of ω-hypergeometric functions in the following theorem.
Proof. Using (1.4), the integral representation for where Using the transformation formulae [1,9], yields where and the proof is complete.

Remark 2.2.
For ω = 1, we get the representation of B in terms of hypergeometric functions given in [11, page 315], that is, . Now we establish the asymptotic expansions for the generalized beta function using the previous theorem. (2.13) Recurrence relations. The following recurrence relations for ω B can be easily derived from its definition and the recurrence relations of ω R given in [12, equations (8)- (12)].
Theorem 2.4. The following relations hold: (2.14) Moreover, using [12, equation (17)] and integration by parts to the integral representation of ω B we obtain the following result.
Lemma 2.5. With the preceding assumptions, one can easily derive We conclude this section by giving the partial derivatives of the generalized beta function. (2.17) Proof. The first formula is obtained by observing that The second formula is obtained by using the integral representation of and recalling that, This ends the proof.

The generalized incomplete beta functions.
We define for x, ω > 0, Re u, Re(a+ µ), Re(b + µ) > 0, and | arg v| < π, the related functions: and call it the generalized incomplete beta function, and its companion function, which may be called the generalized complementary incomplete beta function. In other words, we have The next theorem lists some differential properties and recurrence relations of these incomplete functions. For simplicity, we let Theorem 3.1. Continue with the preceding notations, to obtain the following formulas: (3.7) The proof of the formulas (3.5), (3.6), and (3.7) is straight forward from the respective definitions.

The probability density function.
In a systematic study of generalized pdf and their statistical properties, special functions have played a significant role [2,3,6,7,10]. Chaudhry and Zubair [5,4] have used modified Bessel functions to extend the gamma function, and then used them to define some densities. Kalla et al. [7] have used a generalized form of hypergeometric function to study a new pdf Ismail Ali et al. [3] have used τ-confluent hypergeometric function to define and study a generalized inverse Gaussian distribution. Here we study a new probability density involving the function defined by (1.2).
The pdf of a random variable X associated with (1.2) is defined by, It is obvious that ∞ 0 f (t)dt = 1. We observe that the behavior of f (x) at zero depends on u, that is, Moreover, we have lim x→0 + f (x) = ∞, u < 1, and lim x→∞ f (x) = 0. It can be easily shown that where L = aΓ (c)Γ (b + ω)/Γ (b)Γ (c + ω).  Special cases. If we set ω = 1, the density function becomes Furthermore, if we let b = c, the density function (4.4) reduces to The beta density function of second kind is recovered from (4.1) when a = 0 5. Some statistical function. The aim of this section is to obtain some basic functions associated with the pdf f (x), such as the population moments, the cumulative distribution function (cdf), the survivor function, the hazard rate function, and the mean residue life function.

Population moments.
We derive several types of moments such as the kth moment and the moment generating function. We begin by evaluating the kth moment, since it will be used to obtain the remaining basic moments, such as the mean, the variance and the moment generating function The kth Moment. The kth moment about the origin of the random variable X whose pdf f (x) given by (4.1), is defined by By virtue of (4.1), we get Now since the mean, expected value of the random variable X, is a special case of this moment, namely, the mean is the first moment Similarly, we can obtain the variance of the random variable X, σ 2 X , using (5.2) with k = 2, and (5.3), since it is defined as Moment generating function. The moment generating function of the random variable X is defined by To avoid the difficulty of this integration, we observe, using Taylor expansion, that Using this beside (5.2), we obtain

The distribution function.
The cdf F(x) of the random variable X is given by, hence the survivor function S(x) can be expressed as A particular case of the hazard function h(x) results when ω = 1, that is, Further, for b = c, the hazard function (5.12) reduces to The hazard function of the beta distribution of the second kind is recovered from (5.11) when a = 0 (5.14)

The mean residue life function.
For a random variable X, the mean residue life function is defined by