Abstract
In this paper, we derive the cumulative distribution functions (CDF) and probability density functions (PDF) of the ratio and product of two independent Weibull and Lindley random variables. The moment generating functions (MGF) and the k-moment are driven from the ratio and product cases. In these derivations, we use some special functions, for instance, generalized hypergeometric functions, confluent hypergeometric functions, and the parabolic cylinder functions. Finally, we draw the PDF and CDF in many values of the parameters.
1. Introduction
The distributions of ratio of random variables are widely used in many applied problems of engineering, physics, number theory, order statistics, economics, biology, genetics, medicine, hydrology, psychology, classification, and ranking and selection [1, 2]. Examples include safety factor in engineering, mass to energy ratios in nuclear physics, target to control precipitation in meteorology, inventory ratios in economics, and Mendelian inheritance ratios in genetics [3, 4]. Also, ratio distribution involving two Gaussian random variables is used in computing error and outage probabilities [5]. It has many applications especially in engineering concepts such as structures, deterioration of rocket motors, static fatigue of ceramic components, fatigue failure of aircraft structures, and aging of concrete pressure vessels [6, 7]. An important example of ratios of random variables is the stress-strength model in the context of reliability. It describes the life of a component which has a random strength and is subjected to random stress. The general numerical method is developed for computing the PDF of the sums, products, or ratio of any number of nonnegative independent random variables [8]. The ratio and product distributions have been studied by several authors especially when independent random variables come from the same family or different families. For ratio distribution, the historical review, see [9, 10] for the normal family, [11] for Student’s t family, [12] for the Weibull family, [13] for the noncentral chi-squared family, [14] for the gamma family, [15] for the beta family, [16] for the logistic family, [17] for the Frechet family, [3] for the inverted gamma family, [18] for Laplace family, [7] for the generalized-F family, [19] for the hypoexponential family, [2] for the gamma and Rayleigh families, and [20] for gamma and exponential families. For product distribution, the historical review, see [21] for t and Rayleigh families, [4] for Pareto and Kumaraswamy families, [6] for the t and Bessel families, and [22] for the independent generalized gamma-ratio family. A new distribution is introduced based on compounding Weibull and Lindley distributions; in this approach, several properties of the distribution are derived [23]. In this paper, we derived the ratio and product of cumulative distribution functions (CDF) and probability density functions (PDF) of the independent Weibull and Lindley random variables. In this derivation, we used some special functions and integrals, for instance, generalized and confluent hypergeometric functions, parabolic cylinder function, gamma function of negative integer numbers, and some special integrals. We derived the moment generated function (MGF), m-moment, mean, and variance of the ratio random variable , while we derived the CDF, PDF, and the MGF with respect to the product random variable . The rest of this paper is organized as follows: in Section 2, the derivation of the CDF, PDF, CGF, plots of the CDF and PDF, m-moment, mean, and variance of by using some special functions and integrals are given such that has Weibull distribution and has Lindley distribution. The CDF, PDF, CGF, and plots of the CDF and PDF have been given in Section 3. Finally, some conclusions are considered in Section 4. Weibull distribution with shape parameter and scale parameter , the PDF and CDF are defined as
And the CDF issuch that . Lindley distribution with shape parameter , the PDF and CDF are given as
Also, the CDF issuch that . In this work, we assume that the shape parameter in Weibull distribution. So, the PDF, equation (1), and CDF, equation (2), become, respectively,
The generalized hypergeometric function [24] is defined aswhere Pochhammer symbol or . The confluent hypergeometric function [24] is defined aswhere and . The gamma function of the negative integer numbers [25] is given bywhere and is Euler’s constant [25] such that . The parabolic cylinder function [24] is defined as
The first special integral [26] is given aswhere and . The other special integral [26] is introduced:
The m-moment can be defined as
2. Distribution of the Ratio
In this section, we derive CDF, PDF, MGF, plots of the CDF and PDF, m-moment, mean, and variance of by using some special functions. Let and ; then, it can be found that the CDF of by
By substituting equations (3) and (6) in equation (14), we get
By using the first special integral in equation (11) to calculate the integral and the integral , one can obtain
We substitute equations (16) and (17) in equation (15):
Equation (18) represents the CDF of , where and by the confluent hypergeometric function . We can analysis equation (18) as the following picture:
Equation (19) is CDF of by the generalized hypergeometric function .
The PDF of such that and can be defined as
By substituting equations (3) and (5), one can obtain
By using the first special integral in equation (11) to compute the integrals and , one can obtain
Equation (23) is PDF of , where and by the confluent hypergeometric function. Figure 1 is the plot of CDF in equation (18).
Figure 2 is the plot of PDF in equation (23).
By substituting equation (8) in equation (23), we have
Equation (24) is PDF of -generalized hypergeometric function. We put equation (23) in equation (13) to get the following:
By using the second special integral in equation (12), we obtain as
Equation (27) is m-moment of .
Now, if , the mean of in equation (27) is
If , the second moment in equation (27) is
The variance can be found from equations (28) and (29) as
The MGF of can be calculated aswhere . Using equations (7) and (8) to rewrite equation (31) as the following picture,
To solve the integrals in equation (32), we assume that ; then, equation (32) becomeswhere , , and are gamma functions of the negative integer numbers.
3. Distribution of the Product
In this section, we derive CDF, PDF, MGF, plots of the CDF and PDF, m-moment, mean, and variance of by using some special functions. Let and ; then, the CDF of can be computed by
Equation (33) is CDF of by the confluent hypergeometric function. By using equation (8), equation (33) can be written as the following:
Equation (34) is CDF of by generalized hypergeometric function, where . The PDF of can be considered:where . So,
Figure 3 is the plot of CDF in equation (33). Equation (36) is PDF of by gamma function of the negative integer number.
The MGF of can be computed aswhere . Therefore,
Equation (38) is MGF of .
4. Conclusion
In this paper, we discussed distribution of the ratio independent Weibull and Lindley random variables. In this approach, we derived the CDF, PDF, and MGF of the ratio independent Weibull and Lindley random variables. Also, the plots of CDF and PDF are drawn. The m-moment, mean, and variance are calculated. The CDF and PDF are derived in two formulas with respect to each one of them, first formula by confluent hypergeometric function and another formula by generalized hypergeometric function. We studied distribution of the product independent Weibull and Lindley random variables. In this case, the CDF, PDF, and MGF are derived from the product independent Weibull and Lindley random variables. The plots of CDF and PDF are drawn. The CDF is derived in two formulas: first formula by using confluent hypergeometric function and another formula by using generalized hypergeometric function. However, PDF is derived by using gamma function of negative integer numbers.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors thank every person who helped them to complete this paper.