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On the Product and Quotient of Pareto and Rayleigh Random Variables

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On the Product and Quotient of Pareto and Rayleigh Random Variables Pak. J. Statist. 2019 Vol. 35(4), 285-300 ON THE PRODUCT AND QUOTIENT OF PARETO AND RAYLEIGH RANDOM VARIABLES Noura Obeid and Seifedine Kadry§ Department of Mathematics and Computer Science Faculty of Science, Beirut Arab University, Lebanon §Corresponding author Email: [email protected] ABSTRACT The distributions of products and ratios of random variables are of interest in many areas of the sciences. In this paper, we find analytically the probability distributions of the product XY and the ratio X/Y, when X and Y are two independent random variables following Pareto and Rayleigh distributions respectively. KEYWORDS Product Distribution, Ratio Distribution, Pareto Distribution, Rayleigh Distribution, Error function, probability density function, Moment of order r, Survival function, Hazard function. 1. INTRODUCTION For given random variables X and Y, the distributions of the product XY and the ratio X/Y are of interest in many areas of the sciences. Examples of XY include traditional portfolio selection models, relationship between attitudes and behavior, number of cancer cells in tumor biology and stream flow in hydrology. The distributions of ratio of random variables are of interest in many areas of the sciences, engineering, physics, number theory, order statistics, economics, biology, genetics, medicine, hydrology, psychology, classification, and ranking and selection, see Ali, Nadarajah and Woo (2005), Ali, Pal and Woo (2007), Shcolnick (1985) and Bhargava and Khatri (1981). 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, see Ali, Nadarajah and Woo (2005), Ali, Pal and Woo (2007). Also ratio distribution involving two Gaussian random variables are used in computing error and outage probabilities, see Bu-Salih (1983). 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 the aging of concrete pressure vessels, see Feldstein (1971), Gradshteyn and Ryzhik (2000). 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 Y and is subjected to random stress X. The component fails at the instant that the stress applied to it exceeds the strength and the component will function satisfactorily whenever Y > X. Thus, Pr (X < Y) is a measure of component reliability see Feldstein (1971), Gradshteyn and Ryzhik (2000). The distributions of XY and X/Y have been studied by several © 2019 Pakistan Journal of Statistics 285 286 On the Product and Quotient of Pareto and Rayleigh Random Variables authors, especially when X and Y are independent random variables and come from the same family. With respect to products of random variables, see Sakamoto (1943), Nadarajah (2005), for uniform family, Harter (1951), Steece, B.M. (1976), Wallgren (1980), Nadarajah and Choi (2006) for Student’s t family, Springer and Thompson (1970, Nadarajah and Kotz (2005) for normal family, Stuart (1962), Nadarajah (2007) and Podolski, Nadarajah and Kotz (2006), Podolski (1972) for gamma family, Steece (1976), Nadarajah (2006), Bhargava and Khatri (1981), Harter (1951), Tang and Gupta (1984), Nadarajah (2010) for beta family, AbuSalih (1983), Gradshteyn and Ryzhik (2000), for power function family, and Malik and Trudel (1986), Idrizi (2014) for exponential family (for a comprehensive review of known results, see also Rathie and Rohrer (1987), Modi and Joshi (2012). With respect to ratios of random variables, see Marsaglia (1965), Stuart (1962), and Korhonen and Narula (1989), Tang and Gupta (1984) for normal family, Press (1969), Wallgren (1980) for Student’s t family, Basu and Lochner (1971), Grubel (1968), for Weibull family, Shcolnick (1985) for stable family, Hawkins and Han (1986), Hawkins and Han (1986) for non-central chi-squared family, Provost (1989) for gamma family, and Pham-Gia (2000), Lomnicki (1967) for beta family. In latest years, the study of product and ratio when X and Y belong to different families has been a great interest in many areas of the sciences. For example, distributions of the product and ratio, when X and Y Are Gamma and Weibull independent random variables respectively, have been studied by Nadarajah and Kotz (2006), Park (2010). The distributions of the product and ratio, when X and Y are independent random variables with Pareto and Gamma distributions respectively, have been studied by Nadarajah (2010), Pham-Gia (2000). The Ratio of Pareto and Kumaraswamy Random Variables have been studied by Idrizi (2014), Ali, Pal and Woo (2007). In the applications mentioned before, it is quite possible that X and Y could arise from different but similar distributions. In this paper, we find analytically the probability distributions of the product XY and the ratio X/Y, when X and Y are two independent random variables following Pareto and Rayleigh distributions respectively. with probability density functions (p.d.f.s) 푐푎푐 푓 (푥) = (1) 푋 푥푐+1 푦 −푦2 푓 (푦) = 푒 2푏2 (2) 푌 푏2 respectively, for 푎 ≤ 푥 < ∞, 푎 > 0, 푐 > 0, 0 < 푦 < ∞, 푏 > 0. The calculations of this paper involve several important lemmas LEMMA 1 For 푝 > 0, we have 2 2 2푛 − 1 +∞ 푒−푝 푥 Γ(− , 푝2푢2) 2 2푛−1 (3) ∫ 2푛 푑푥 = 푝 푢 푥 2 Proof: ( ) +∞ 푠−1 −푡 By definition we know that Γ s, x = ∫푥 푡 푒 푑푡 if we substitute 푡 = 푝2푥2 in (3) we get Noura Obeid and Seifedine Kadry 287 2푛 +∞ −푡 2푛−1 +∞ 2푛+1 푝 푒 푝 푝 −( ) 2 −푡 2 ∫ 푛 1⁄2 푑푡 = ∫ 푡 푒 푑푡 2푝 푝2푢2 푡 푡 2 푝2푢2 2푛 − 1 (4) Γ(− , 푝2푢2) = 2 푝2푛−1 2 LEMMA 2 For 푝 > 0, n = 0,1, … , we have +∞ 2푛 −푝푥2 (2푛 − 1)!! 휋 ∫ 푥 푒 푑푥 = 푛 √ (5) 0 2(2푝) 푝 Proof: First of all, we have +∞ 2 ∫ 푒−푥 푑푥 = √휋 (6) −∞ Let 푋 = √푝푥 +∞ +∞ 2 1 2 ∫ 푥2푛푒−푝푥 푑푥 = ∫ 푋2푛푒−푋 푑푋 2푛 0 (√푝) √푝 0 +∞ (7) 1 2 = ∫ 푋2푛푒−푋 푑푋 푛 푝 √푝 0 +∞ 2 (2푛)!√휋 Let 푎 = ∫ 푋2푛푒−푋 푑푋, prove that 푎 = 푛 0 푛 22푛푛!2 Integration by parts implies: +∞ (2푛 − 1) 2푛−2 −푋2 (2푛 − 1)2푛 푎푛 = ∫ 푋 푒 푑푋 = 2 푎푛−1 (8) 2 0 2 푛 휋 And since 푎 = √ we conclude 0 2 +∞ 2 (2푛)! √휋 2푛 −푋 푎푛 = ∫ 푋 푒 푑푋 = 2푛 0 2 푛! 2 and +∞ 2 (2푛)! √휋 ∫ 푥2푛푒−푝푥 푑푥 = (9) 푛 2푛 0 푝 √푝2 푛! 2 1 (2푛)! (2푛−1)‼ And since Γ (푛 + ) = √휋 = √휋 2 4푛푛! 2푛 1 +∞ Γ(푛 + ) 2 2 (2푛 − 1)‼ 휋 ∫ 푥2푛푒−푝푥 푑푥 = = √ (10) 푛 푛 0 2푝 √푝 2(2푝) 푝 288 On the Product and Quotient of Pareto and Rayleigh Random Variables LEMMA 3 For 푝 > 0 +∞ 2푛+1 −푝푥2 푛! ∫ 푥 푒 푑푥 = 푛+1 (11) 0 2푝 Proof: If we substitute 푢 = 푝푥2 we get +∞ +∞ 2푛+1 −푝푥2 1 푛 −푢 ∫ 푥 푒 푑푥 = 푛+1 ∫ 푢 푒 푑푢 (12) 0 2푝 0 To get the result we have to prove +∞ ∫ 푢푛푒−푢푑푢 = 푛! (13) 0 For 푛 = 0 integration by parts yields: +∞ ∫ 푒−푢푑푢 = 1 = 0! (14) 0 Assume the statement holds for some arbitrary natural number 푛 ≥ 0, And prove that the statement holds for 푛 + 1. Integration by parts implies: +∞ +∞ ∫ 푢푛+1푒−푢푑푢 = (푛 + 1) ∫ 푒−푢푢푛푑푢 = (푛 + 1)푛! = (푛 + 1)! (15) 0 0 2. DISTRIBUTION OF THE PRODUCT 푿풀 Theorem 2.1. Suppose 푋 and 푌 are independent and distributed according to (1) and (2), respectively. Then for 푧 > 0 the cumulative distribution function c.d.f. of 푍 = 푋푌 can be expressed as: Case 1: If 푐 + 1 = 2푛, 푎 푐 1 푐‼ 퐹 (푧) = 1 − ( ) [ √2푏2휋] (16) 푍 푧 푏2 1 푐+1 2( ) 2 푏2 Case 2: If 푐 + 1 = 2푛 + 1 Noura Obeid and Seifedine Kadry 289 푐 1 ( ) ! 퐹 (푧) = 1 − [ 2 ] (17) 푍 푏2 1 푐+2 2( ) 2 2푏2 Proof: 푎 The c.d.f. corresponding to (1) is 퐹 (푥) = 1 − ( )푐 Thus, one can write the c.d.f. of 푋 푥 푋푌 as: +∞ 푧 Pr(푋푌 ≤ 푧) = ∫ 퐹푋 ( ) 푓푌(푦)푑푦 0 푦 +∞ 푐 푦2 푎푦 푦 − 2푏2 = ∫ [1 − ( ) ] 2 푒 푑푦 (18) 0 푧 푏 푐 +∞ 푦2 푎 1 − 푐+1 2푏2 = 1 − ( ) 2 ∫ 푦 푒 푑푦 푧 푏 0 where, 푧 > 0, 푎 > 0, 푐 > 0, 푦 > 0, 푏 > 0. The proof of Theorem 2.1 easily follows by using Lemma 2 when 푐 + 1 = 2푛, and Lemma 3 when 푐 + 1 = 2푛 + 1 in the integral above. Corollary 2.2. Let 푋 and 푌 are independent and distributed according to (1) and (2), respectively. Then for 푧 > 0, the probability density function p.d.f. of 푍 = 푋푌 can be expressed as: Case 1: If 푐 + 1 = 2푛, 푐 푐푎 1 푐‼ 푓 (푧) = √2푏2휋 (19) 푍 푧푐+1 푏2 푐+1 1 2 2 ( ) [ 푏2 ] Case 2: If 푐 + 1 = 2푛 + 1 푐 푐 ( ) ! 푐푎 1 2 푓 (푧) = (20) 푍 푧푐+1 푏2 푐+2 1 2 2 ( ) [ 2푏2 ] For 푧 > 0. Proof: The probability density function 푓푍(푧) in (20) and (19) easily follows by differentiation. 290 On the Product and Quotient of Pareto and Rayleigh Random Variables Corollary 2.3 Let 푋 and 푌 are independent and distributed according to (1) and (2), respectively. Then for 푐 > 푟 and 푧 ≥ 훼, 훼 > 0. the moment of order 푟 of 푍 = 푋푌 can be expressed as: Case 1: If 푐 + 1 = 2푛, 푟−푐 훼 1 푐‼ 퐸[푍푟] = 푐푎푐 √2푏2휋 (21) 푐 − 푟 푏2 푐+1 1 2 2 ( ) [ 푏2 ] Case 2: If 푐 + 1 = 2푛 + 1 푐 푟−푐 ( ) ! 훼 1 2 퐸[푍푟] = 푐푎푐 (22) 푐 − 푟 푏2 푐+2 1 2 2 ( ) [ 2푏2 ] Proof: Case 1: If 푐 + 1 = 2푛, +∞ +∞ 푐 푐푎 1 푐‼ 퐸[푍푟] = ∫ 푧푟푓 (푧)푑푧 = ∫ 푧푟 √2푏2휋 푑푧 푍 푧푐+1 푏2 푐+1 −∞ 훼 1 2 [2 ( 2) ] 푏 (23) 푟−푐 훼 1 푐‼ = 푐푎푐 √2푏2휋 푐 − 푟 푏2 푐+1 1 2 2 ( ) [ 푏2 ] For 푐 > 푟, 푧 ≥ 훼, 훼 > 0 Same proof for case 2.
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