Distributions of the Ratio and Product of Two Independent Weibull and Lindley Random Variables

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. )e 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.


Introduction
e 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. e general numerical method is developed for computing the PDF of the sums, products, or ratio of any number of nonnegative independent random variables [8]. e 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 gammaratio 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 Z � X/Y, while we derived the CDF, PDF, and the MGF with respect to the product random variable XY. e 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 Z � X/Y by using some special functions and integrals are given such that X has Weibull distribution and Y has Lindley distribution. e 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 X ∼ W(a, b) with shape parameter a > 0 and scale parameter b > 0, the PDF and CDF are defined as And the CDF is such that X > 0. Lindley distribution Y ∼ L(c) with shape parameter c > 0, the PDF and CDF are given as Also, the CDF is such that Y > 0. In this work, we assume that the shape parameter a � 2 in Weibull distribution. So, the PDF, equation (1), and CDF, equation (2), become, respectively, e generalized hypergeometric function [24] is defined as where Pochhammer symbol (d) l � (Γ(d + l))/Γ(d) or (e) l � (Γ(e + l))/Γ(e). e confluent hypergeometric function [24] is defined as where w ∈ R and e ≠ 0, ± 1, ± 2, ± 3, . . .. e gamma function of the negative integer numbers [25] is given by where ϕ(r) � r l�1 1/l and η � lim n⟶∞ n l�1 ((1)/(l− ln(n))) is Euler's constant [25] such that Γ(0) � − η. e parabolic cylinder function [24] is defined as e first special integral [26] is given as where Re(σ) > 0 and Re(ρ) > 0. e other special integral [26] is introduced: . (12) e m-moment can be defined as

Distribution of the Ratio Z = � X/Y
In this section, we derive CDF, PDF, MGF, plots of the CDF and PDF, m-moment, mean, and variance of Z � X/Y by using some special functions. Let X ∼ W (2, b) and Y ∼ L(c); then, it can be found that the CDF of Z � X/Y by By substituting equations (3) and (6) in equation (14), we get 2 Journal of Probability and Statistics By using the first special integral in equation (11) to calculate the integral (c 2 )/(c + 1) We substitute equations (16) and (17) in equation (15): Equation (18) represents the CDF of Z � X/Y, where X ∼ W(2, b) and Y ∼ L(c) by the confluent hypergeometric function Ψ(d, e; w). We can analysis equation (18) as the following picture: Equation (19) is CDF of Z by the generalized hypergeometric function p Υ q (d 1 , d 2 , . . . , d p ; e 1 , e 2 , . . . , e q ; w). e PDF of Z � X/Y such that X ∼ W(2, b) and Y ∼ L(c) 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 Equation (23) is PDF of Z � X/Y, where X ∼ W(2, b) and Y ∼ L(c) 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 z-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 Z. Now, if m � 1, the mean of Z in equation (27) is If m � 2, the second moment in equation (27) is e variance V(Z) can be found from equations (28) and (29) as (29) e MGF of Z can be calculated as where ω � (b 2 c 2 )/(4z 2 ). Using equations (7) and (8) to rewrite equation (31) as the following picture, To solve the integrals in equation (32), we assume that − R � (cbω (− 1)/2 t)/(2); then, equation (32) becomes

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. e m-moment, mean, and variance are calculated. e 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. e plots of CDF and PDF are drawn. e 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
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.