Review of the Air Force Academy No.2 (37)/2018 THE RAYLEIGH’S FAMILY OF DISTRIBUTIONS Bogdan Gheorghe MUNTEANU ”Henri Coandă” Air Force Academy, Braşov, Romania ([email protected]) DOI: 10.19062/1842-9238.2018.16.2.3 Abstract: Several general mathematical properties of the Rayleigh′s family of distributions are examined in a consistent manner by using the power series distributions (PSD) class [1]. A new cumulative distribution function and probability density are obtained for the continuous type random variables which represent the maximum or the minimum in a sequence of independent, identically Rayleigh distributed random variables, in a random number by means of a power series distribution. An asymptotic result characterized by the Poisson Limit Theorem is formulated and analysed. Keywords: power series distributions, Rayleigh distribution, distribution of the maximum and minimum, Poisson Limit Theorem 1. INTRODUCTION In the paper [2] sets out to introduce and analyse the properties of the maximum and minimum distributions for a sample of power series distribution. This can serve as a mathematical model to describe the probabilistic behaviour of the signals used on a large scale in the field of radiolocation. In this paper, the distribution is presented as being the distribution of the maximum or minimum value from a sample of random volume Z from a Rayleigh distributed statistical population, where Z is a random value from the power series distribution class. 2. MIN RAYLEIGH AND MAX RAYLEIGH POWER SERIES DISTRIBUTIONS It is know that a random variable admits a Rayleigh distribution with the parameter , and we note X: Rayleigh( ), 0 , if the cumulative distribution function (cdf) is x2 2 2 FRay ( x ) 1 e , x 0 , while the corresponding probability density function (pdf) x2 x 2 f( x ) e2 , x 0 . Ray 2 We consider the random variables UXXXRay max 12 , ,..., Z and VXXXRay min 12 , ,..., Z , where Xi i1 are independent and identically distributed random a z variables, X: Rayleigh ( ), 0 and Z PSD , that is P(Z z) z , z 1,2,..., where i A( ) aa12, ,... is a sequence of real, non-negative numbers. 0 the radius of convergence of the z power series Aa() z , (0, ) and the real parameter of the distribution. z1 31 The Rayleigh’s Family of Distributions We point out that the random variables are independent of the random variable Z , the latter’s distribution being part of the power series distributions class [1]. In accordance with the working methods in the paper [2,4], it can be stated that the random variables U Ray follow the Max Rayleigh power series distributions of parameters and (we note: URay : MaxRayleighPS (,) ) and VRay follow the Min Rayleigh power series distributions of parameters and (we note: VRay : MinRayleighPS (,) ) if the cumulative distribution functions (cdf) are characterized by the relation: x2 Ae1 2 2 A FRay () x U( x ) , x 0 (1) Ray AA()() and x2 A() A e 2 2 A( ) A 1 FRay ( x ) V( x ) , x 0. (2) Ray AA()() The probability densities functions (pdf) are characterized by the relation: xx22 22d d xe22 A 1 e f()() x A F x dx Ray Ray u( x )dx , x 0 , (3) Ray AA()() 2 and, xx22 22d d x e22 A e f( x ) A 1 F ( x ) Ray Ray dx v( x )dx , x 0. (4) Ray AA()() 2 Proposition 2.1. If is a sequence of independent random variables, Rayleigh distributed with parameters 0 , while where with , , then: k x2 2 2 * limURay ( x ) 1 e , x 0, where kmin k N , ak 0 . 0 UXXX max , ,..., Proposition 2.2. If is a sequence of independentRay random variables,12 Rayleigh Z Xi i1 distributed with parameters , while VXXXRay max 12 , ,..., Z where a z Z PSD P(Z z) z , z 1,2,..., with , , then: A( ) xl2 2 2 z * limVRay ( x ) 1 Aa e() , x 0, where l(0,min ) l N , al 0. 0 z z1 32 Review of the Air Force Academy No.2 (37)/2018 Corollary 2.1. The rth moments, rN , r 1 of the random variables and are given by: z r r az EURay Emax X12 , X ,..., X z z1 A() and URay : MaxRayleighPS (,) z r r az VRay : MinRayleighPS (,) EVRay Emin X12 , X ,..., X z , z1 A() where the pdfs of the random variables maxXXX12 , ,..., z and minXXX12 , ,..., z z1 z1 are f()()() x z f x F x and f( x ) z f ( x ) 1 F ( x ) . max XXX12 , ,..., z Ray Ray min XXX12 , ,..., z Ray Ray 3. SPECIAL CASES 3.1. The Max Rayleigh Binomial and Min Rayleigh Binomial distributions. The Max Rayleigh Binomial (MaxRayB) and Min Rayleigh Poisson (MinRayP) distributions are defined by the distribution functions presented in a general framework in [2], where n p A( ) 1 1, with , p(0,1): 1 p n x2 1 e 2 2 1 1 A FRay () x Ux() RayB A( ) ( 1)n 1 (5) n x2 1p e2 2 (1 p )n = ,x 0 1 (1p )n and n x2 11n e 2 2 A1 FRay ( x ) Vx( ) 1 RayB A( ) ( 1)n 1 (6) n x2 11 p p e 2 2 = ,x 0. 1 (1p )n respectively. 3.2. The Max Rayleigh Poisson and Min Rayleigh Poisson distributions. The Max Rayleigh Poisson (MaxRayP) and Min Rayleigh Poisson (MinRayP) distributions are characterized by the cumulative distributions functions defined by the relations (1) and * (2), where Ae(* ) 1, with * , 0 : * * FxRay () A FRay () x e 1 Ux() RayP A()* * e 1 (7) x2 2 eee 2 = ,x 0 1 e 33 The Rayleigh’s Family of Distributions and x2 2 1 e 2 * A1 FRay ( x ) 1 e V( x ) 1 = , x 0. (8) RayP Ae(* ) 1 3.3. On the Poisson limit theorem. The following theorems show that the MaxRayP and MinRayP distributions approximate the MaxRayB and MinRayB distributions depending on certain conditions. Theorem 3.1. (Poisson limit theorem). The MaxRayP and MinRayP distributions can be obtained as the limit of the MaxRayB, respectively MinRayB distributions with distribution functions given by (5) and (6) if n when n and 0 . In other words, ( i) limVRayB ( x ) V RayP ( x ), x 0 , where VxRayB () and VRayP ( x ), x 0 are the n p0 distribution functions of the random variables VRayB : MinRayleighB(,,) n p and VRayP : MinRayleighPoi(,) defined by (6) and (8); (ii) limURayB ( x ) U RayP ( x ), x 0, where UxRayB () and URayP ( x ), x 0 are t n p0 distribution functions of the random variables URayB : MaxRayleighB(,,) n p and URayP : MaxRayleighPoi(,) defined by (5) and (7). Proof. We examine the convergence in terms of the maximum distributions UxRayB () and UxRayP (), x 0 . It is evident that: np lim (1p )n lim 1 p1/ p e ; nn pp00 22 np e x /2 x22/2 2 2 2 2 1/ pe lim (1p ex/ 2 ) n lim 1 p e x / 2 nn pp00 22 lim np e x /2 n x22/2 eep0 e . 22n xn/2 x22/2 1pe (1 p ) ee e limURayB ( x ) lim U RayP ( x ), x 0. nn 1 (1 pe )n 1 pp00 Fig. 1 and 2 show the behaviour of the pdfs of MinRayleighB(,,) n p , MinRayleighPoi (,), MaxRayleighB(,,) n p and MaxRayleighPoi(,) for some 1 values of the parameters: n 40 , p , 4 , 5 . 10 Fig. 3 and 4 show the behaviour of the pdfs of , , and for some values of the parameters. 34 Review of the Air Force Academy No.2 (37)/2018 1 1 cdfMaxRayB40 5x 10 0.8 cdfMaxRayPoi(45x) 1 cdfMaxRayB40 4x0.6 10 cdfMaxRayPoi(44x) 1 0.4 cdfMaxRayB40 3x 10 cdfMaxRayPoi(43x) 0.2 Cumulative distributionCumulative function 0 0 5 10 15 20 x Fig. 1: Pdfs for the Max-Rayleigh-Binomial and Max-Rayleigh-Poisson distributions – graphical illustration of the Poisson Limit Theorem 0.3 1 pdfMaxRayB40 5x 10 pdfMaxRayPoi(45x) 0.2 1 pdfMaxRayB40 4x 10 pdfMaxRayPoi(44x) 1 pdfMaxRayB40 3x0.1 10 Probability Probability density function pdfMaxRayPoi(43x) 0 0 5 10 15 20 x Fig. 2: Pdfs for the Max-Rayleigh-Binomial and Max-Rayleigh-Poisson distributions – graphical illustration of the Poisson Limit Theorem 1 1 cdfMinRayB40 5x 10 0.8 cdfMinRayPoi(45x) 1 cdfMinRayB40 4x0.6 10 cdfMinRayPoi(44x) 0.4 1 cdfMinRayB40 3x 10 cdfMinRayPoi(43x) 0.2 Cumulative distributionCumulative function 0 0 5 10 15 20 x Fig. 3: Pdfs for the Min-Rayleigh-Binomial and Min-Rayleigh-Poisson distributions – graphical illustration of the Poisson Limit Theorem 35 The Rayleigh’s Family of Distributions 0.4 1 pdfMinRayB40 5x 10 0.3 pdfMinRayPoi(45x) 1 pdfMinRayB40 4x 10 0.2 pdfMinRayPoi(44x) 1 pdfMinRayB40 3x 10 0.1 Probability Probability density function pdfMinRayPoi(43x) 0 0 5 10 15 20 x Fig. 4: Pdfs for the Min-Rayleigh-Binomial and Min-Rayleigh-Poisson distributions – graphical illustration of the Poisson Limit Theorem CONCLUSIONS The results formulated and examined in this paper are in connection with the study of the random variable distribution, which can be expressed as being the maximum or minimum of a sequence of independent random variables identically distributed in a random number. In practice, this translates as the emission and reception of some signals is a random number, signals whose amplitude is a random variable characterized by the Rayleigh distribution [3]. The signals that best record either a maximum or minimum amplitude are of special interest to us. It has, thus, been presented in a consistent manner how to determine the maximum and minimum distribution of independent and identically distributed random variables, which form a random sequence.