The Rayleigh's Family of Distributions

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 i1 are independent and identically distributed random a  z variables, X: Rayleigh ( ), 0 and ZPSD , 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. z1

31 The Rayleigh’s Family of Distributions

We point out that the random variables Xi i1 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  Ae1 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 UXXXRay max 12 , ,..., Z  where ZPSD z az z with P(Z  z)  , z  1,2,..., Aa()  z  ,  (0, ) , then: A( ) z1 k x2  2 2 * limURay ( x ) 1  e , x  0, where kmin k  N , ak  0 . 0    Proposition 2.2. If is a sequence of independent random variables, Rayleigh distributed with parameters , while VXXXRay max 12 , ,..., Z  where with , , then:

xl2  2 2 * limVRay ( x ) 1  e , x  0, where lmin l  N , al  0. 0

32 Review of the Air Force Academy No.2 (37)/2018

Corollary 2.1. The rth moments, rN , r 1 of the random variables

URay : MaxRayleighPS(,)  and VRay : MinRayleighPS(,)  are given by: z r r az EURay  Emax X12 , X ,..., X z  z1 A() and z r r az EVRay  Emin X12 , X ,..., X z  , z1 A() where the pdfs of the random variables maxXXX12 , ,..., z  and minXXX12 , ,..., z  z1 z1 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  1p  e2 2  (1  p )n  = ,x  0 1 (1p )n and n x2   11n  e 2 2   A1 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 UxRayP ()* A()* e 1 (7)

x2  2 eee 2  = ,x  0 1 e

33 The Rayleigh’s Family of Distributions and

x2  2  1 e 2 *   A1 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 p0 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 p0 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 (1p )n  lim 1  p1/ p  e ; nn   pp00

22 np e x /2  x22/2 2 2 2 2 1/ pe lim (1p  ex/ 2 ) n  lim 1  p  e x / 2 nn    pp00

22 lim np e x /2 n  x22/2 eep0  e .

22n xn/2  x22/2 1pe  (1  p ) ee e   limURayB ( x ) lim   U RayP ( x ), x  0. nn  1 (1 pe )n 1   pp00 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  cdfMaxRayB40 5x  10 0.8

cdfMaxRayPoi(45x)  1  cdfMaxRayB40 4x0.6  10 

cdfMaxRayPoi(44x)  1 0.4 cdfMaxRayB40 3x  10 

cdfMaxRayPoi(43x) 0.2 Cumulative distribution Cumulative 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  pdfMaxRayB40 5x  10 

pdfMaxRayPoi(45x) 0.2  1  pdfMaxRayB40 4x  10 

pdfMaxRayPoi(44x)  1  pdfMaxRayB40 3x0.1  10 

Probability Probability density function pdfMaxRayPoi(43x)

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  cdfMinRayB40 5x  10 0.8

cdfMinRayPoi(45x)  1  cdfMinRayB40 4x0.6  10 

cdfMinRayPoi(44x) 0.4  1  cdfMinRayB40 3x  10 

cdfMinRayPoi(43x) 0.2 Cumulative distribution Cumulative 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  pdfMinRayB40 5x  10  0.3 pdfMinRayPoi(45x)  1  pdfMinRayB40 4x  10  0.2 pdfMinRayPoi(44x)  1  pdfMinRayB40 3x  10  0.1

Probability Probability density function pdfMinRayPoi(43x)

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. The Poisson Limit Theorem has been formulated when the random variable number in a sequence has a zero truncated binomial distribution and the limit distribution of the minimum and maximum is of Poisson type.

REFERENCES

[1] N.L. Johnson , A. W. Kemp, S. Kotz, Univariate Discrete Distribution, New Jersey, 2005; [2] A. Leahu, B. Gh. Munteanu, S. Cataranciuc, On the lifetime as the maximum or minimum of the sample with power series distributed size, Romai J., vol. 9, no. 2, pp. 119-128, 2013; [3] B. R. Mahafza, Radar systems analysis and design using Matlab, CRC Press Taylor & Francis Group, USA, 2013; [4] B. Gh. Munteanu, The Min-Pareto power series distributions of lifetime, Appl. Math. Inf. Sci., vol. 10, no. 5, pp. 1673-1679, 2016; [5] B. Gh. Munteanu, Qualitative aspects of the Min Pareto binomial distribution, Review of the Air Force Academy, vol. XV, no.2(34), pp. 63-68, 2017.