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Product Of n Independent Maxwell Random Variables
Noura Obeid, Seifedine Kadry
Department of Mathematics and Computer Science, Faculty of Science, Beirut Arab University, Lebanon [email protected], [email protected]
Abstract
We derive the exact probability density functions (pdf) of a product of n independent Maxwell distributed random variables. The distribution functions are derived by using an inverse Mellin transform technique from statistics, and are given in terms of a special function of mathematical physics, the Meijer G-function.
Keywords: Product Distribution, Maxwell Distribution, Mellin transform technique, Meijer G-function, probability density function.
1 Introduction
Engineering, Physics, Economics, Order statistics, Classification, Ranking, Selection, Number theory, Genetics, Biology, Medicine, Hydrology, Psychology, these all applied problems depend on the distribution of product of random variables[1][2].
As an example of use of the product of random variables in physics, Sornette [27] mentions: "...To mimic system size limitation, Takayasu, Sato, and Takayasu introduced a threshold xc ...and found a stretched exponential truncating the power-law pdf beyond xc. Frisch and Sornette recently developed a theory of extreme deviations generalizing the central limit theorem which, when applied to multiplication of random variables, predicts the generic presence of stretched exponential
pdfs. The problem thus boils down to determining the tail of the pdf for a product of random variables
//
Several authors have studied the product distributions for independent random variables come from the same family or different families, 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. In this paper, we find analytically the probability distributions of the product n"=1 , when Xi is a Maxwell random variable with probability density function (p.d.f)
The functions are derived by using an inverse Mellin transform technique from statistics and given in terms of the Meijer G-function.
2
-X
2
2 Basic Definitions
2.1 Mellin integral transform
The Mellin integral transform of f(x) is defined only for x >0, as:
M{f(x)/s} = E[xs-1] = J™ xs-1f(x)dx (2)
The inverse transform is:
f(x) = x-sM[f(x)/s}ds (3)
The path of integration is any line parallel to the imaginary axis and lying within the strip of analyticity of M{f(x)/s}.
The Mellin integral transform of the density function f(x) of the product X = X1.X2...Xn of n independentrandom variables Xi with the densityfunction fx.(xi) is defined as:
M[fx(x)/s} = nn=i M{fxt(xi)/s} (4)
Using the inverse transform formula we obtain the density function of the product distribution as:
fx(x) = ^ Cr x- nn=i M{fxt(xi)/s}ds (5)
2.2 Meijer G-function
The Meijer G-function is defined by the contour integral:
Gpq(Zlbi.....bq ) = 2jn^c-rZ ^=n+1T(s+almlm+1T(1-bl-s)dS (6)
where z, {ai}i, and{bi}l are in general, complex-valued. The contour is chosen so that it separates the poles of the gamma products in the numerator. The Meijer G-function has been implemented in some commercial mathematics software packages.
3 Product of n Independent Maxwell Random Variables
Theorem 1: Suppose Xi, i = 1,..,n are independent random variables distributed according to (1). Then for x > 0 the probability density function p.d.f. of X = n7n=1 Xi can be expressed as:
fx(x) = 2($nnntb;Gnn(x22-nnn=1 b-\.....i ) (7)
ProofConsider a product of n independent random variables
X = n=1 Xi (8)
where Xi is a Maxwell distributed random variable with probability density function according to (1), The Mellin integral transform of fx.(xi) is:
M[fxl(xi)/s} = J™ xf-1fXi(xi)dxi
= W^Cxre2(bi)2dxi (9)
= ^2s/21(b-2)Ml + s/2)
Where we have used the definition of the gamma function
m = 0 xt-1e-x (10)
The Mellin integral transform of fx(x)
M{fx(x)/s} = n=1 M{fxl(xi)/s}
= n=1 [J22s/2±i( b-2yh(1 + s/2)] We can find the pdf of X as the inverse Mellin transform
2
-X
1 c°+i
fx(x) =7ZS,
2 j nJc-]~ 1 rc+jx
х-х[Пи
ъ ф5/2Ь(ь-
:)-2Г(1 +s/2)]]ds
= 2in£j- (*2)~фп(2-пг
П?=1ь,
(пи (ьд-2)-2т=1 r(i+^)2f
(ii)
= К12У
1 rc+J
:fc
VLi=1bi2jnJc-i
Finally using the definition of the Meijer G-function we get
(х22-пП=1 Ь-2)-2П=1 r(1+f)f
fx(x) = KB
Фпж^Мх22~
i,-,i
(12)
Corollary 1: Suppose Xi, i = l,..,n are independent random variables distributed according to (1). Then for t > 0 the cumulative distribution function c.d.f. of X = n"}=1 Xi can be expressed as:
Fx(t) = 2 (Jz)n \ Gln+1 ( t22-nn=1 b-2\2
К nf=1bi2'
2
(13)
Proof The cumulative distribution function Fx(t) = f fx(x)dx is obtained by integrating (7) with respect to x inside the contour integral by using:
f x-sdx = — = t1-s 1 (—) = t1-s 1(1- -)-1
J0 1-s 2 2 2 2
And
Then we get
Let ßn = 2(£)n 1
15 15
1 S _
2 2 Ti--5-)
22
f x- d x = J0
1
1 Щ-д
1-S 1 K2 2 3 s
2 rp—) 22
К
F
(t) = tßn^Cß 1 (t2)-^(2-n ПП=1 b-2)-**^®^^
Finally using the definition of the Meijer G-function we obtain
Fx(t) = KB
nJ П?=1^2
— rn1
- U1n+1 (
t 2-n ПП=1 b-2\
1 2|2
1,...,1,-~
2
(14)
(15)
Corollary 2: Suppose Xi, i = l,..,n are independent random variables distributed according
to (1). Then for r > 0,a > 0 the moment of order r of X = n""^ Xi can be expressed as:
Proof
We have Then
Also we have
EVI = 2(12)
n Пi=lЬi2
1 rn1 — U
1 r -2i2 2
1n+1 ( a 2 n ПЧ=1 b- \2 r 1
1,...,1,----
22
E[Xr] =f+™xrfx(x)dx
r + rn
f + xrfx(x)dx
ßn^Ci- fa:xr-(2-nun=1 K2Ts/2пП=1 r(1 + s-)dxf
Г
Ja
d x =
\ — r —
1 rc+a1
Е[ХГ] = ßn~~fc-jx
s\ ds
s —/ —
■(2-n П=1 b- a2)-s/2 П=1 Г(1 +
(16)
(17)
(18)
s
s
1
1
1
n
a
n
1+T—s
a
x
1 1 r^+^+i)
s-r-1 2 r(—+-+3) 2 2 2'
Finally using (19) and the definition of the Meijer G-function we obtain
1 1 r-1- S r(—+-+i) c
E[Xr] = ßn^ar+1(-±) CI- (2-n nu ^^r-lrtinu r(1 +
( 2 2 2)
1 r
= 2(^)n Gnn+i («22-n n=i b-2 _r__!_
(19)
(20)
22
Corollary 3: Suppose Xi, i = 1,..,n are independent random variables distributed according to (1). Then for a > 0 the expected value of X = n7}=1 Xi can be expressed as: For r = 1
E[X] = z^r-^-lcrl+i (a22-n nn=i b-2l0.....^ ) (21)
Corollary 4: Suppose Xi, i = 1,..,n are independent random variables distributed according to (1). Then for a > 0 the expected value of X = ni}=1 Xi can be expressed as:
= ^ni^^+Ja^mi b-\lx_3
(22)
—
-[2(J^)nu£ri2Cinn1+i(a22-nnn=i b-2l°°.....1,-1 )]2
Proof. By definition the variance of X/Y is:
a2 =E[Z2]-E[Z]2 (23)
Corollary 5:Suppose Xi, i = 1,..,n are independent random variables distributed according to (1). Then for x > 0 the survival function of X = n}=1 Xi can be expressed as:
/1 if x < 0
Sx(x) =
y n'i=1bi2-in+iy - "i=1~i '1>_>1>-
Proof By definition of the survival function
Sx(x) = 1 - Fx(x) (25)
Corollary 6:Suppose Xi, i = 1,..,n are independent random variables distributed according to (1). Then for x > 0 the hazard function of X = n}=1 Xi can be expressed as:
/0 ifx < 0
^njrf^-^X.....i ) ifx > 0 (26)
1-Klbn7ihr5c?1+i\x22-nnn=iK2\l ^ 1 ) ifx > 0 (24)
hx(x) =
Ir.m I v2n-nr,n h-2!2
= 1ui2 \ 1.....1 —
a2
4 Examples and special cases
4.1 Product of two independent Maxwell random variables
1. Probability density function: Suppose Xi, i = 1,2 are independent Maxwell random variables with scale parameters b1 = 1,b2 = 2 respectively, the probability density function of X is
/0 ifx < 0
fx(x) = \2-G2°(^U ) ifx>0 (27)
2. Cumulative distribution function: Suppose Xi, i = 1,2 are independent Maxwell random variables with scale parameters b1 = 1,b2 = 2 respectively, the cumulative distribution function of X is For t > 0
(0 ift<1
1
Fx(t) =
^(S^ ) *>o (28)
3. Moment of order "r": Suppose Xi, i = 1,2 are independent Maxwell random variables with scale parameters b1 = 1,b2 = 2 respectively, the moment of order r of X is For a > 0
E[Xr] = -a—G2i(-\2ii2_r__i_ I (29)
'' 2 2
4. Expected value: Suppose Xi, i = 1,2 are independent Maxwell random variables with scale parameters b1 = 1,b2 = 2 respectively, the Expected value of X is
zW = -aicli1(?6\°1X-1 ) (30)
5. Variance: Suppose Xi, i = 1,2 are independent Maxwell random variables with scale parameters b1 = 1,b2 = 2 respectively, the Variance of X is
r+1 „2
3 ¡2
= --Gin-\ 2 _ " 16 1,1,
(31)
-[-aiGH(f/1X-1 )]
6. Survival function: Suppose Xi, i = 1,2 are independent Maxwell random variables with scale parameters b1 = 1,b2 = 2 respectively, the Survival function of X is
/1 ift < 0
sx(t) =
1
t-2
V-iGSlS^ I — (32)
7. Hazard function: Suppose Xi, i = 1,2 are independent Maxwell random variables with scale parameters b1 = 1,b2 = 2 respectively, the Hazard function of X is For > 0
/0 ifx < 0
hx{x) =
lr20(Hl I
n02\lb\l,1
1 £r2ll
fx > 0 (33)
1 r
1
a2
2
Figure 1: Plot of the probability density function for two independent Maxwell random variables
for b1 = 1,b2 = 2.
Figure 2: Plot of the cumulative distribution function for two independent Maxwell random
variables for b1 = l,b2 = 2.
Figure 3: Plot of the hazard function for two independent Maxwell random variables for b1 = 1,b2 = 2.
5 Applications
The air molecules surrounding us are not all traveling at the same speed, even if the air is all at a single temperature. Some of the air molecules will be moving extremely fast, some will be moving with moderate speeds, and some of the air molecules will hardly be moving at all. Because of this, we can't ask questions like "What is the speed of an air molecule in a gas?" since a molecule in a gas could have any one of a huge number of possible speeds.
So instead of asking about any one particular gas molecule, we ask questions like, "What is the distribution of speeds in a gas at a certain temperature?" In the mid to late 1800s, James Clerk Maxwell and Ludwig Boltzmann figured out the answer to this question. Their result is referred to as the Maxwell-Boltzmann distribution, because it shows how the speeds of molecules are distributed for an ideal gas. The Maxwell-Boltzmann distribution is often represented with the following graph.
The y-axis of the Maxwell-Boltzmann graph can be thought of as giving the number of molecules per unit speed. So, if the graph is higher in a given region, it means that there are more gas molecules moving with those speeds.
Let take the following example: we are interested to find the distribution X = X1X2X3X4XSX6X7X8X9X10, where Xi are independent Maxwell random variables with scale parameters bi,b1 = l,b2 = 2, b3 = 3,b4 = 4, b5 = 5,b6 = 6,b7 = 7,b8= Q,b9 = 9,b10 = 10.
So the speeds of molecules are distributed for an ideal gas with respect to the probability density function of X
Maxwell-B o ltzmann
distribution
speed iJ
Figure 4: Maxwell-Boltzmann distribution
fxW = 2(^)10
(12345678910)
1
Go100 (X22-1O(
(34)
PDF 1.« 10"®
8.* 10"10
6.* 10_1tJ
4.« 10"1[J
2.*10"'c
0 5.0 *1O0 1.0* 10? 1.5 H 10s 20*10^
Figure 5: Plot of n=10 independent Maxwell rondom variables for b1 = 1 ,b2 = 2, b3 = 3,b4 =
4, b5 =5 ,b6 = 6,b7 = 7,b8 = 8 ,b9 = 9 ,b10 = 10.
6 Monte Carlo simulation:
Monte Carlo simulations are used to model the probability of different outcomes in a process that cannot easily be predicted due to the intervention of random variables. It is a technique used to understand the impact of risk and uncertainty in prediction and forecasting models.
A Monte Carlo simulation can be used to tackle a range of problems in virtually every field such as finance, engineering, supply chain, and science. It is also referred to as a multiple probability simulation.
0.15
o.io
0Д5
оло- ■
0 5 10 15
Figure 6: Monte Carlo simulation for the product of two independent maxwell random variables for scale parameters b1 = 1 ,b2 = 2.
7 Conclusion
This paper has derived the analytical expressions of the PDF, CDF, the moment of order , the survival function, and the hazard function, for the distribution of X = n"i=1 Xi when Xi are Maxwell random variables distributed independently of each other, we have illustrated our results for n = 2 as a special case, then we have discussed an application of the distribution of product X = n"i=1 Xi, finally, we have confirmed our result using Monte Carlo simulation.
References
[1] S. Nadarajah, D. Choi, Arnold and StraussB^™s bivariate exponential distribution products and ratios, New Zealand Journal of Mathematics, 35 (2006), 189-199.
[2] M. Shakil, B. M. G. Kibria, Exact distribution of the ratio of gamma and Rayleigh random variables, Pakistan Journal of Statistics and Operation Research, 2 (2006), 87-98.
[3] M. M. Ali, M. Pal, and J. Woo, On the ratio of inverted gamma variates, Austrian Journal of Statistic, 36 (2007), 153-159.
[4] L. Idrizi, On the product and ratio of Pareto and Kumaraswamy random variables, Mathematical Theory and Modeling, 4 ( 2014), 136-146.
[5] S. Park, On the distribution functions of ratios involving Gaussian random variables, ETRI Journal, 32 (2010), 6.
[6] S. Nadarajah and S. Kotz, On the product and ratio of t and Bessel random variables, Bulletin of the Institute of Mathematics Academia Sinica, 2 (2007), 55-66.
[7] T. Pham-Gia, N. Turkkan, Operations on the generalized-fvariables and applications, Statistics, 36 (2002), 195-209.
[8] G. Beylkin, L. MonzRn, and I. Satkauskas, On computing distributions of products of non-negative independent random variables, Applied and Computational Harmonic Analysis, 46 (2019), 400-416.
[9] P. J. Korhonen, S. C. Narula, The probability distribution of the ratio of the absolute values of two normal variables, Journal of Statistical Computation and Simulation, 33 (1989), 173-182.
[10] G. Marsaglia, Ratios of normal variables and ratios of sums of uniform variables, Journal of the American Statistical Association, 60 (1965), 193-204.
[11] S. J. Press, The t-ratio distribution, Journal of the American Statistical Association, 64 (1969), 242-252.
[12] A. P. Basu and R. H. Lochner, On the distribution of the ratio of two random variables having generalized life distributions, Technometrics, 13 (1971), 281-287.
[13] D. L. Hawkins and C.-P. Han, Bivariate distributions of some ratios of independent noncentral chi-square random variables, Communications in Statistics - Theory and Methods, 15 (1986), 261-277.
[14] S. B. Provost, On the distribution of the ratio of powers of sums of gamma random variables, Pakistan Journal Statistics, 5 (1989), 157-174.
[15] T. Pham-Gia, Distributions of the ratios of independent beta variables and applications, Communications in Statisticse^"Theory and Methods, 29 (2000), 2693-2715.
[16] S. Nadarajah and A. K. Gupta, On the ratio of logistic random variables, Computational Statistics and Data Analysis, 50 (2006), 1206-1219.
[17] S. Nadarajah and S. Kotz, On the ratio of frr©chet random variables, Quality and Quantity, 40 (2006), 861-868.
[18] S. Nadarajah, The linear combination, product and ratio of Laplace random variables, Statistics, 41 (2007), 535-545.
[19] K. Therrar and S. Khaled, The exact distribution of the ratio of two independent hypoexponential random variables, British Journal of Mathematics and Computer Science, 4 (2014), 2665-2675.
[20] L. Joshi and K. Modi, On the distribution of ratio of gamma and three parameter exponentiated exponential random variables, Indian Journal of Statistics and Application, 3 (2014), 772-783.
[21] K. Modi and L. Joshi, On the distribution of product and ratio of t and Rayleigh random variables, Journal of the Calcutta Mathematical Society, 8 (2012), 53-60.
[22] C. A. Coelho and J. T. Mexia, On the distribution of the product and ratio of independent generalized gamma-ratio, Sankhya: The Indian Journal of Statistics, 69 (2007), 221-255.
[23] A. Asgharzadeh, S. Nadarajah, and F. Sharafi, Weibull lindley distributions, Statistical Journal, 16 (2018), 87-113.
[24] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series, Gordon and Breach Science Publishers, Amsterdam, Netherlands, 2 (1986).
[25] F. Brian and K. Adem, Some results on the gamma function for negative integers, Applied Mathematics and Information Sciences, 6 (2012), 173-176.
[26] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, Cambridge, MA, USA, 6 (2000).
[27] D. Sornette Multiplicative processes and power law, Physical Review E, 57 (1998), 4811-4813.
[28] N. Obeid, S. Kadry, On the product and quotient of pareto and rayleigh random variables, PJS Headquarters Lahore, (2019).
