The Distribution of the Ratio of the Products of Two Independent<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:math>-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:math>Variates and Its Application in the Performance Analysis of Relaying Communication Systems

Matovic, Ana; Mekić, Edis; Sekulović, Nikola M.; Stefanović, Mihajlo; Matovic, Marija; Stefanović, Časlav

doi:10.1155/2013/147106

Cited by 13 publications

(11 citation statements)

References 18 publications

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“…Determining distributions of the functions of random variables is a very crucial task and this problem has been attracted a number of researchers because there are numerous applications in Risk Management, Finance, Economics, Science, and, many other areas, see, for example, (Donahue 1964;Ly et al 2016;Nadarajah and Espejo 2006;Springer 1979). Basically, the distributions of an algebraic combination of random variables including the sum, product, and quotient are focused on some common distributions along with the assumptions of independence or correlated through Pearson's coefficient or dependence via multivariate normal joint distributions (Arnold and Brockett 1992;Bithas et al 2007;Cedilnik et al 2004;Hinkley 1969;Macalos and Arcede 2015;Marsaglia 1965;Matović et al 2013;Mekićet al 2012;Nadarajah and Espejo 2006;Kotz 2006a, 2006b;Pham-Gia et al 2006;Pham-Gia 2000;Rathie et al 2016;Sakamoto 1943). Regarding ratio, it often appears in the problems of constructing statistics used in hypothesis testing and estimating issues.…”

Section: Introductionmentioning

confidence: 99%

Determining Distribution for the Quotients of Dependent and Independent Random Variables by Using Copulas

Pho

et al. 2019

JRFM

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Determining distributions of the functions of random variables is a very important problem with a wide range of applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient Y = X 1 X 2 and the ratio of one variable over the sum of two variables Z = X 1 X 1 + X 2 of two dependent or independent random variables X 1 and X 2 by using copulas to capture the structures between X 1 and X 2 . Thereafter, we extend the theory by establishing the density and distribution functions for the quotients Y = X 1 X 2 and Z = X 1 X 1 + X 2 of two dependent normal random variables X 1 and X 2 in the case of Gaussian copulas. We then develop the theory on the median for the ratios of both Y and Z on two normal random variables X 1 and X 2 . Furthermore, we extend the result of median for Z to a larger family of symmetric distributions and symmetric copulas of X 1 and X 2 . Our results are the foundation of any further study that relies on the density and cumulative probability functions of ratios for two dependent or independent random variables. Since the densities and distributions of the ratios of both Y and Z are in terms of integrals and are very complicated, their exact forms cannot be obtained. To circumvent the difficulty, this paper introduces the Monte Carlo algorithm, numerical analysis, and graphical approach to efficiently compute the complicated integrals and study the behaviors of density and distribution. We illustrate our proposed approaches by using a simulation study with ratios of normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas. We find that copulas make big impacts from different Copulas on behavior of distributions, especially on median, spread, scale and skewness effects. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. The approaches developed in this paper are flexible and have a wide range of applications for both symmetric and non-symmetric distributions and also for both skewed and non-skewed copulas with absolutely continuous random variables that could contain a negative range, for instance, generalized skewed-t distribution and skewed-t Copulas. Thus, our findings are useful for academics, practitioners, and policy makers.

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Section: Introductionmentioning

confidence: 99%

Determining Distribution for the Quotients of Dependent and Independent Random Variables by Using Copulas

Pho

et al. 2019

JRFM

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“…These parameters are m 1 , m 2 and κ. When m 1 =1, m 2 =1 and κ=0, the Nakagami-m* Nakagami-m/Rician distribution reduces to Rayleigh*Rayleigh/Rayleigh distribution, when m 1 =1, m 2 =1 the Nakagami-m*Nakagami-m/Rician distribution becomes Rayleigh*Rayleigh/Rician distribution, when κ=0, the Nakagami-m*Nakagami-m/ Rician distribution becomes Nakagami-m*Nakagamim/Rayleigh distribution and for m 1 =1 the Nakagami-m *Nakagami-m/Rician distribution reduces to Rayleigh*Nakagami-m/Rician distribution [8], [9], [10], [11]. To the best of the authors', knowledge radio relay system with two sections performance analyses in the presence Nakagami-m short term fading and Rician co-channel interference is not reported in open technical literature.…”

Section: Introductionmentioning

confidence: 99%

Radio Relay System With Two-Section Performance in the Presence of Nakagami-M Fading and Rician Co-Channel Interference

Milosavljević¹,

Milić²,

Milosavljević³

et al. 2018

Safety Engineering

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Wireless mobile relay communication system with two sections operating over short term fading channel in the presence of co-channel interference subjected to small scale fading has been considered. Desired signal experiences Nakagami-m fading and co- channel interference experiences Rician fading. Desired signal at output of the second section has Nakagami-m*Nakagami-m distribution. The signal-to-interference ratio of product of two Nakagami-m random variables and Rician random variable has been presented. In this paper, closed form expresions for moments of signal- to-interference ratio of considered system are calculated. The influence of Nakagami-m short term fading severity parameter and Rician κ factor on presence is analysed and discussed. Mobile relay communication system with two sections can lead to a more reliable and efficient transmission, which is of great importance in the risky environments. Key words: Nakagami-m fading, Rician fading, moment, mobile relay communications system

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“…improvement of wireless system performances using reliable transmission techniques can be of crucial importance [5]. Further, multi-hop amplify-andforward (AF) relay system signal envelope can be modeled as the product of two or more random variables where the first random variable is the signal envelope at the first section, the second variable is the signal envelope at the second section while the last random variable is the signal envelope at the last section [6,7]. Multi-hop relay systems can be applied in different scenario to increase safety such as: car-tocar (C2C) and car-to-everything (C2X) communications, and to enable reliable connections via drones [6].…”

Section: Introductionmentioning

confidence: 99%

“…Further, multi-hop amplify-andforward (AF) relay system signal envelope can be modeled as the product of two or more random variables where the first random variable is the signal envelope at the first section, the second variable is the signal envelope at the second section while the last random variable is the signal envelope at the last section [6,7]. Multi-hop relay systems can be applied in different scenario to increase safety such as: car-tocar (C2C) and car-to-everything (C2X) communications, and to enable reliable connections via drones [6]. Moreover, amplify-and-forward (AF) multihop relays beside its reliability and efficiency are simple for practical realization since they don't require decoding and other signal processing.…”

Section: Introductionmentioning

confidence: 99%

Application of Laplace Approximation Formula in Performance Analysis of Wireless Relay Communication System in Multipath Fading Environment

Stefanović¹

2017

Safety Engineering

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In this paper, Laplace approximation formula is efficiently used to calculate infinite-series expression for average level crossing rate (LCR) of the product of Nakagami-m and kappa-mu random processes. The results can be used in performance analysis of dual-hop relay wireless mobile communication system in specific multipath fading environment when the first section is under the influence of NLOS multipath environment while the second section is under the influence of LOS multipath fading environment. The influences of multipath fading parameters on the LCR of the proposed model are examined, graphically presented and discussed. Moreover, analytical approach of applying Laplace approximation formula in multi-hop systems is further considered by obtaining closed form expressions for the LCR of the product of three and four Nakagami-m processes, respectively. Key words: average level crossing rate, kappa-mu multipath fading, Laplace approximation formula, Nakagami-m distribution, relay communications system.

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The Distribution of the Ratio of the Products of Two Independentα-μVariates and Its Application in the Performance Analysis of Relaying Communication Systems

Cited by 13 publications

References 18 publications

Determining Distribution for the Quotients of Dependent and Independent Random Variables by Using Copulas

Determining Distribution for the Quotients of Dependent and Independent Random Variables by Using Copulas

Radio Relay System With Two-Section Performance in the Presence of Nakagami-M Fading and Rician Co-Channel Interference

Application of Laplace Approximation Formula in Performance Analysis of Wireless Relay Communication System in Multipath Fading Environment

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