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provided by Tomsk State University Repository Advances in Intelligent Systems Research (AISR), volume 151 2018 International Conference on Computer Modeling, Simulation and Algorithm (CMSA 2018)

The Statistical Properties of the Probabilistic Distributions Generated by Gamma Statistics

Oleg V. Chernoyarov1,2, Alexandra V. Salnikova2,3* and Vladimir I. Kostylev4 1Russia, 111250, Moscow, Krasnokazarmennaya st., 14, National Research University “Moscow Power Engineering Institute” 2Russia, 634050, Tomsk, Lenin Avenue, 36, International Laboratory of Statistics of Stochastic Processes and Quantitative Finance of National Research Tomsk State University 3Russia, 394026, Voronezh, Moscow Avenue, 14, Voronezh State Technical University 4Russia, 394018, Voronezh, Universitetskaya sq., 1, Voronezh State University *Corresponding author

Abstract—We show that hyperexponential and hyper- which is under integer values of the distributions reproduce themselves by composition. We find new second parameter. The sums of squared absolute values of formulas for probability densities and distribution functions of independent complex Gaussian variables (statistics [1]) with Gamma statistics sums. zero mean values and identical dispersions obey the Erlang distribution. Thus, the Erlang distribution is also frequently Keywords—composition of statistics; replicating statistics; found in the reliability theory [1, 2]. exponential, Gamma, Erlang distribution. Generalization of the is the I. INTRODUCTION hyperexponential distribution [2] with the probability density Gamma distribution [1-3] is one of the most common probabilistic distributions in the statistical radio physics and   M radio engineering [4, 5]. Particularly, by means of Gamma  e ,,; aMxW  ;axw mem   distribution, we can describe the effects at the outputs of the m1 typical signal processing systems receiving the discrete Gaussian processes [5], multifrequency signals [6], radiation where M is an integer parameter of the hyperexponential from complex radio sources [7], etc. The statistical properties distribution, while  ,,,  and  21 M of Gamma distribution can be described by means of the   ,,, aaaa  are real parameters of the probability density 21 M hyperexponential distribution. Similarly, the hyper Gamma and hyper-Erlang distributions can be introduced with the AA  1 probability densities    1,; exp AaxxaxAaxw      M where a is the first [3] (scale [1]) parameter, A is the second [3]  ,,,; AaMxW  m  ,; Aaxw mm and 1 , x  0 , m1 (mean [1]) parameter of Gamma distribution; 1x   0 , x  0  M  A1   is the Heaviside function;  exp   dyyyA is the  E ,,,; KaMxW  ,; Kaxw mmEm   0  m1 Gamma function [8]. Generally, both parameters of Gamma distribution are real.  correspondingly. Here  21 ,,, AAAA M and The special case (under A  1 ) of Gamma distributions is  the exponential distribution. The probability density of the   21 ,,, KKKK M . exponential distribution has the form of [2] It is well known that some statistical distributions (such as normal, chi-square, binomial, Poisson) reproduce themselves  exp1;   aaxxaxw   by composition. Gamma distribution has a reproducing e property by the second parameter only [3]: if two independent Gamma statistics have identical first parameters, then their sum As another special case, the Erlang distribution can be also has Gamma distribution. The aim of the present paper is to introduced with the probability density [3] define the statistical properties of the sums of two independent Gamma statistics under arbitrary parameter values as well as KK  1 some sums related to them.  E  1,; expKaxxaxKaxw  1 !  

Copyright © 2018, the Authors. Published by Atlantis Press. 258 This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). Advances in Intelligent Systems Research (AISR), volume 151

II. EXPONENTIAL STATISTICS We compare Eqs. (5) and (8) and see that the sum of independent Erlang and exponential statistics obeys hyper- The probability density ;,baxw of the sum of two ee  Erlang distribution. The distribution function corresponding to independent exponential statistics with the parameters a and b the probability density (8) takes the form of can be found as the convolution of two probability densities of the form (2). As a result, we obtain K Ee  1,,; abbbKaxf    ;,;  ;  abbxbwbaaxawbaxw   K l 1 ee e e   K   ab    ab       ;  abxbw       ,; laxw  . e  b b E We then compare Eqs. (4) and (6) and see that the sum of  l 1       the two independent exponential statistics has the hyperexponential distribution. It is simple to show also that the We can see that the formulas (6) and (7) can be obtained from distribution function Eqs. (8) and (9) as a special case while K  1 . Based on Eqs. (8) and (9), it is easy to write the expressions 2 2 for the probability density and the distribution function of the  ee  e ;1,;  e ;   abbxwbbaaxwabaxf   sum of hyper-Erlang and hyperexponential statistics. corresponds to the probability density (6). V. GAMMA AND EXPONENTIAL STATISTICS

III. HYPEREXPONENTIAL STATISTICS The probability density of the sum of independent Gamma statistics with the parameters a and A and exponential statistics It is obvious that the probability density     with the parameter b can be found as the convolution of the ee   ,,,,,; bNaMxW  of the sum of two independent probability densities (1) and (2) and it has the form of   hyperexponential statistics with the parameters ,, aM and   ,, bN is the hyperexponential probability density, namely, A    b   ab   e ,,; bAaxw    e   xfbxw ,;; A     ab    ba      M N ee ,,,,,; bNaMxW  ,; baxw nmeenm  Here        ,P1,; axAxAaxf  is the distribution function m11n corresponding to the probability density (1), M N z 1   ; mem   ;bxwBaxwA nen , ,P zt  t1 exp  dyyy is the normalized incomplete m1 n1  t  0 Gamma function [8]. N  M  where  aA n and  bB m . mmm   ba nnn   ab The distribution function corresponding to Eq. (10) can be n1 nm m1 mn presented in the form of Thus, it can be argued that the set of the hyperexponential probability densities is closed with respect to the convolution operation. That means that hyperexponential probability  e      ,;,,;  e ,,; bAaxbwAaxfbAaxf   distribution replicates itself by composition. while the probability density and the distribution function of IV. ERLANG AND EXPONENTIAL STATISTICS the sum of independent hyper Gamma statistics with    By calculating the convolution of the probability densities parameters   ,,, AaM  and hyperexponential statistics with   (2) and (3), we find the expression for the probability density the parameters  ,, bN  have the appearance Ee ,,; bKaxW of the sum of independent Erlang and exponential statistics that appear to have the form of      e  ,,,,,,; bNAaMxW  A Ee ,,; bKaxW  N M m  bn   ba nm  K l  ;bxw     xf ,; A  ,   b   a K   ab    mnen   ab     ab m      ;bxw    ,; laxw  , n11m  mn   mn   ab e  ba  b E    l 1    where a and K are the Erlang statistics parameters, b is the exponential statistics parameter.

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     l e  ,,,,,,; bNAaMxF    ab  1 K l1 l ,,, LbKaI    ,  M  N    b  2  LK l1  m   ,; Aaxf mm   ,,; bAaxwb nmmenn  . m11 n  l   ba  1 L ,,, LbKaJ    l1 VI. ERLANG STATISTICS l   a  2  LK l1 From [3], we know the expression for the characteristic function of the Erlang distribution and  m   1  maaaa 1 is the Pochhammer symbol [8]. K  E 1,;  aiKa   By applying the linear property of the Fourier transform, from Eq. (17), we can easily obtain the general expression for the probability density of the sum of two independent Erlang where i  1 is the imaginary unit. statistics: The characteristic function of the sum of two independent statistics with the parameters a, K and b, L is the product of two characteristic functions of the form (13), that is, EE   ,,,,,,; LbKaCLbKaxw   K L  (18)   l ,,, E ,;   l ,,, E ,; lbxwLbKaJlaxwLbKaI  , K L    EE  11,,,;  biaiLbKa    l1 l1 

In a specific case, from Eq. (14), we get We compare Eqs. (5) and (18) and see that the sum of two independent Erlang statistics obeys the hyper-Erlang distribution. From this it follows that the sum of two   1,,,; aiLaKa   LK  ,;  LKa  independent hyper-Erlang statistics also has the hyper-Erlang EE E distribution, that is, the hyper-Erlang distribution reproduces itself by composition. which means that the Erlang distribution is reproducible by the second parameter. From Eq. (18), it is easy to obtain the expression for the distribution function of the sums of two independent Erlang Generally, the right-hand member of Eq. (14) can be statistics: presented in the form of the sum of common fractions [9], namely,

EE   ,,,,,,; LbKaCLbKaxf  K L  ,,,; LbKa    EE   l ,,, E ,;   l ,,, E ,; lbxfLbKaJlaxfLbKaI  , KL K l  l 1 l 1  L ba lLK  1 !   ab   1    LK    1 !  abL l 1  lK  1!  aib  where KL L l K ba lLK  1 !   ba   1 .  LK    K 1 l 1 !  baK l 1  lL  1!  bia   x  1  x    Kaxf exp1,;       E  la ! a (16)   l0   We now transform Eq. (16) into is the Erlang distribution function corresponding to the probability density (3). If we take into account Eq. (19), the EE   ,,,,,,; LbKaCLbKa  expression (18) can be transformed into  K L  (17)   l ,,, E ,;   l ,,, E  ,; lbLbKaJlaLbKaI  , l 1 l 1  where the designation are

L K  a   b  LK  2 ! ,,, LbKaC      ,   ba    ab  1 ! LK  1 !

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As F     10,, irrespective of α and β, from Eq. (22) it  11 EE 1,,,;   ,,,1 LbKaCxLbKaxf  follows that  m   x  K l 1 1  x     ,;,,,;  BAaxwBaAaxw   exp  ,,, LbKaI      a l m! a    l 1 m0   Implication of Eq. (23) is analogous to the meaning of Eq. (15): m  x  L l 1 1  x    under  ba (and only in this case) the sum of two independent ,,, exp  ,,, LbKaJ     . b l m! b Gamma statistics has Gamma distribution.   l 1 m0     For the distribution function of the sum of two independent Gamma statistics, we managed to get the integral expression VII. GAMMA AND ERLANG STATISTICS only, in the form of Let there be two independent statistics, one of which described by Gamma distribution with the parameters a, A and 1 the other – by Erlang distribution with the parameters b, B. ,,,; BbAaxf  ,; Aaxw  Then the probability density of their sum is the convolution of  a  the probability densities (1) and (3). After simple, but though  1    xy  cumbersome, transformations we obtain 1 A1 Bbxyfy exp,;   dy .   b 0  

L  a  It should be noted that the limits of integration and the ,,,; LbAaxw    ,; LAaxw  E b  subintegral function in Eq. (24) are such that the numerical      computation by this formula is not difficult.  x x  11  LALF ,,   ,  a b  IX. CONCLUSION Thus, we have found the new expressions for the sums of   z m two independent random variables which obey Gamma, where ,, zF  m is the confluent 11   m! exponential or Erlang distribution. We can show that the set of m0 m hyperexponential probability densities, as well as the set of hypergeometric function (Kummer's function) [8]. hyper-Erlang probability densities, are closed with respect to The distribution function corresponding to Eq. (20) can be the convolution operation, while the set of hyper Gamma presented in the form of the finite sum: probability densities does not possess this property.

CKNOWLEDGMENT L A  E   ,;,,,;   E ,,,; lbAaxwbAaxfLbAaxf   This study was financially supported by the Russian l1 Science Foundation (research project No. 17-11-01049).

In the specified case, when L  1 , we obtain the expression REFERENCES (10) from Eq. (20) as well as the expression (11) from Eq. (21). [1] W. Feller, An Introduction to Probability Theory and Its Applications, Thus, the expression for the distribution function of the sum of Vol. 2. New York, Wiley, 1971. hyper Gamma and hyper-Erlang statistics generalizes the [2] V.S. Korolyuk, N.I. Portenko, A.V. Skorokhod and A.F. Turbin, formula (12). Handbook on Probability Theory and Mathematical Statistics [in Russian]. Moscow, Mir, 1984. VIII. GAMMA STATISTICS [3] K.O. Bowman and L.R. Shenton, Properties of Estimators for the Gamma Distribution. New York, Marcel Dekker, 1988. The probability density of the sum of two independent [4] H.L. van Trees, K.L. Bell and Z. Tian, Detection, Estimation, and Gamma statistics is determined by the formula similar to Eq. Modulation Theory. Part I. Detection, Estimation and Filtering Theory. (20), namely, New York, Wiley, 2013. [5] B.R. Levin, Theoretical Principles of Radioengineering Statistics. Virginia, Defense Technical Information Center, 1968. B [6] P.A. Bakut, I.A. Bolshakov, G.P. Tartakovsky, et. al., Problems of the  a  Statistical Theory of Radar. Virginia, Defense Technical Information  ,,,; BbAaxw     ,; BAaxw  b Center, 1964.      [7] V.I. Kostylev, “Detection of extended radio source in noise”,  x x  Radiophysics and Quantum Electronics, vol. 42, pp. 352-356, April 1999. 11  BABF ,,   .  a b  [8] Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Edited by M. Abramowitz and I. A. Stegun, National Bureau of Standards, Applied Mathematics Series 55, USA, 1964.

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[9] G.A. Korn and T.M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York, Dover Publications, 2000.

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