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Journal articles on the topic 'Distribution (probability theory)'

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Published: 4 June 2021

Last updated: 31 July 2024

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1

Tarasov, Vasily E. "Nonlocal Probability Theory: General Fractional Calculus Approach." Mathematics 10, no. 20 (October 17, 2022): 3848. http://dx.doi.org/10.3390/math10203848.

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Nonlocal generalization of the standard (classical) probability theory of a continuous distribution on a positive semi-axis is proposed. An approach to the formulation of a nonlocal generalization of the standard probability theory based on the use of the general fractional calculus in the Luchko form is proposed. Some basic concepts of the nonlocal probability theory are proposed, including nonlocal (general fractional) generalizations of probability density, cumulative distribution functions, probability, average values, and characteristic functions. Nonlocality is described by the pairs of Sonin kernels that belong to the Luchko set. Properties of the general fractional probability density function and the general fractional cumulative distribution function are described. The truncated GF probability density function, truncated GF cumulative distribution function, and truncated GF average values are defined. Examples of the general fractional (GF) probability distributions, the corresponding probability density functions, and cumulative distribution functions are described. Nonlocal (general fractional) distributions are described, including generalizations of uniform, degenerate, and exponential type distributions; distributions with the Mittag-Leffler, power law, Prabhakar, Kilbas–Saigo functions; and distributions that are described as convolutions of the operator kernels and standard probability density.

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2

Fang, Zizhou, Kaixi Tan, and Ziyi Wang. "Fundamental results in probability theory." Highlights in Science, Engineering and Technology 49 (May 21, 2023): 464–69. http://dx.doi.org/10.54097/hset.v49i.8586.

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Probability theory is an area of mathematics that deals with the concept of likelihood. Probability theory is the mathematical foundation of statistical reasoning, and understanding how unpredictability impacts data is crucial for data scientists. Gaussian (normal) distribution is the most widely used distribution. It has two parameters which are mean and variance and easy to interpret. Also, the central limit theorem tells us that sums of independent random variables make the least number of assumptions. In addition, Poisson, Laplace, Beta, Pareto, Dirichelt, Binomial and Gamma Distributions are useful in different areas. The multivariate Gaussian is the most widely used joint probability density function. Covariance and correlation are used to measure the degree between two random variable’s X and Y. Chebyshev Inequality defines a topological space, which includes a sequence of elements, and let the sequence be called . Strong Law of Large Numbers Theorem use in large number of random variable in pairwise independent identically distributed and Renewal Theory is and example in Strong Law of Large Numbers Theorem.

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3

Zeina, Mohamed Bisher, Nizar Altounji, Mohammad Abobala, and Yasin Karmouta. "Introduction to Symbolic 2-Plithogenic Probability Theory." Galoitica: Journal of Mathematical Structures and Applications 7, no. 2 (2023): 18–30. http://dx.doi.org/10.54216/gjmsa.070202.

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In this paper we present for the first time the concept of symbolic plithogenic random variables and study its properties including expected value and variance. We build the plithogenic formal form of two important distributions that are exponential and uniform distributions. We find its probability density function and cumulative distribution function in its plithogenic form. We also derived its expected values and variance and the formulas of its random numbers generating. We finally present the fundamental form of plithogenic probability density and cumulative distribution functions. All the theorems were proved depending on algebraic approach using isomorphisms. This paper can be considered the base of symbolic plithogenic probability theory.

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4

Tarasov, Vasily E. "Fractional Probability Theory of Arbitrary Order." Fractal and Fractional 7, no. 2 (February 1, 2023): 137. http://dx.doi.org/10.3390/fractalfract7020137.

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A generalization of probability theory is proposed by using the Riemann–Liouville fractional integrals and the Caputo and Riemann–Liouville fractional derivatives of arbitrary (non-integer and integer) orders. The definition of the fractional probability density function (fractional PDF) is proposed. The basic properties of the fractional PDF are proven. The definition of the fractional cumulative distribution function (fractional CDF) is also suggested, and the basic properties of these functions are also proven. It is proven that the proposed fractional cumulative distribution functions generate unique probability spaces that are interpreted as spaces of a fractional probability theory of arbitrary order. Various examples of the distributions of the fractional probability of arbitrary order, which are defined on finite intervals of the real line, are suggested.

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5

Taylor, G. C. "A Heuristic Review of some Ruin Theory Results." ASTIN Bulletin 15, no. 2 (November 1985): 73–88. http://dx.doi.org/10.2143/ast.15.2.2015020.

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AbstractThe paper deals with the renewal equation governing the infinite-time ruin probability. It is emphasized as intended to be no more than a pleasant ramble through a few scattered results. An interesting connection between ruin probability and a recursion formula for computation of the aggregate claims distribution is noted and discussed. The relation between danger of the claim size distribution and ruin probability is reexamined in the light of some recent results on stochastic dominance. Finally, suggestions are made as to the way in which the formula for ruin probability leads easily to conclusions about the effect on that probability of the long-tailedness of the claim size distribution. Stable distributions, in particular, are examined.

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6

ROY, S. M., and VIRENDRA SINGH. "CAUSAL QUANTUM MECHANICS TREATING POSITION AND MOMENTUM SYMMETRICALLY." Modern Physics Letters A 10, no. 08 (March 14, 1995): 709–16. http://dx.doi.org/10.1142/s0217732395000752.

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De Broglie and Bohm formulated a causal quantum mechanics with a phase space density whose integral over momentum reproduces the position probability density of the usual statistical quantum theory. We propose a causal quantum theory with a joint probability distribution such that the separate probability distributions for position and momentum agree with the usual quantum theory. Unlike the Wigner distribution the suggested distribution is positive-definite and obeys the Liouville condition.

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7

V. P. Singh, L. Zhang, and A. Rahimi. "Probability Distribution of Rainfall-Runoff Using Entropy Theory." Transactions of the ASABE 55, no. 5 (2012): 1733–44. http://dx.doi.org/10.13031/2013.42364.

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8

Kondratiev, Yuri G. "WHITE NOISE DISTRIBUTION THEORY (Probability and Stochastics Series)." Bulletin of the London Mathematical Society 32, no. 1 (January 2000): 119–20. http://dx.doi.org/10.1112/s0024609399306391.

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9

Bougoffa, Lazhar, and Panagiotis T. Krasopoulos. "Integral inequalities in probability theory revisited." Mathematical Gazette 105, no. 563 (June 21, 2021): 263–70. http://dx.doi.org/10.1017/mag.2021.56.

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10

Tillé, Yves. "Yet Another Attempt to Classify Positive Univariate Probability Distributions." Austrian Journal of Statistics 53, no. 3 (June 10, 2024): 87–101. http://dx.doi.org/10.17713/ajs.v53i3.1776.

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We propose an original classification of several discrete and continuous probability distributions. We establish links between these distributions, in particular the little known relationship between the negative hypergeometric distribution and the beta distribution. These relations allow us to propose a structure of relations which is summarized in graphic form.Our classification emphasises the analogy between certain discrete and continuous distributions. This analogy makes it possible to establish relations between the theory of point processes and the theory of survey sampling. It also makes it possible to envisage the use of link functions that are little used in generalised regression.

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11

I.I., Itopa, Isa A.M., and Bashiru S.O. "Transmuted Topp-Leone Exponential Distribution: Theory and Application to Real Dataset." African Journal of Mathematics and Statistics Studies 6, no. 2 (April 8, 2023): 80–88. http://dx.doi.org/10.52589/ajmss-j9fsnj5r.

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The main aim of this study is to add to the existing literature on probability distributions. In this study, the transmutation map approach proposed by Shaw and Buckley (2007) was used to develop a probability distribution called Transmuted Topp-Leone Exponential (TTLE) distribution. The moment, moment generating function and entropy are among the statistical properties of the distribution that were derived. The maximum likelihood approach was used to estimate the parameters of the novel distribution. The TTLE distribution was applied to a real-world data set and compared to other well-known standard distributions; the result of the analysis revealed that the newly developed distribution is more superior than the competing models.

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12

Li, Wen, and Michael K. Ng. "On the limiting probability distribution of a transition probability tensor." Linear and Multilinear Algebra 62, no. 3 (March 19, 2013): 362–85. http://dx.doi.org/10.1080/03081087.2013.777436.

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13

Li, Yan, Wei Xiao, Hai Yang Huang, and Ying Cai Yuan. "Total-Probability Method in Reliability Design of Gear Strength." Applied Mechanics and Materials 37-38 (November 2010): 1604–14. http://dx.doi.org/10.4028/www.scientific.net/amm.37-38.1604.

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Total probability method is based on stress-strength interference theory. There are four acceptable distribution types for gear (strength, stress), which are normal distribution, lognormal distribution, Weibull distribution and the smallest extreme value distribution. Based on the four distribution types, 16 different stress-strength distributions were obtained through permutation respectively. The results show that they summarize the mathematical model of reliability calculation for the gear actual failure mode and get the exact solutions of gear reliability.

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14

Chen, Jincan, Tie Liu, Zhifu Huang, and Guozhen Su. "Probability distribution function of complex systems." International Journal of Modern Physics B 32, no. 03 (January 22, 2018): 1850022. http://dx.doi.org/10.1142/s0217979218500224.

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Based on the probability distribution observed in some complex systems and an assumption that the entropies of complex systems satisfy a pseudo additivity, it is expounded that a new probability distribution function is suitable for not only a single complex system but also a coupling complex system, and consequently, a statistical theory of complex systems can be established in the extensive-like framework.

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15

Badr, Majdah, and Muhammad Ijaz. "The Exponentiated Exponential Burr XII distribution: Theory and application to lifetime and simulated data." PLOS ONE 16, no. 3 (March 26, 2021): e0248873. http://dx.doi.org/10.1371/journal.pone.0248873.

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The paper addresses a new four-parameter probability distribution called the Exponentiated Exponential Burr XII or abbreviated as EE-BXII. We derive various statistical properties in addition to the parameter estimation, moments, and asymptotic confidence bounds. We estimate the precision of the maximum likelihood estimators via a simulation study. Furthermore, the utility of the proposed distribution is evaluated by using two lifetime data sets and the results are compared with other existing probability distributions. The results clarify that the proposed distribution provides a better fit to these data sets as compared to the existing probability distributions.

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16

Rajeev, S. G. "A Theory Of Errors in Quantum Measurement." Modern Physics Letters A 18, no. 33n35 (November 20, 2003): 2439–50. http://dx.doi.org/10.1142/s0217732303012672.

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It is common to model the random errors in a classical measurement by the normal (Gaussian) distribution, because of the central limit theorem. In the quantum theory, the analogous hypothesis is that the matrix elements of the error in an observable are distributed normally. We obtain the probability distribution this implies for the outcome of a measurement, exactly for the case of traceless 2 × 2 matrices and in the steepest descent approximation in general. Due to the phenomenon of 'level repulsion', the probability distributions obtained are quite different from the Gaussian.

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17

Hannig, J., and J. S. Marron. "Advanced Distribution Theory for SiZer." Journal of the American Statistical Association 101, no. 474 (June 1, 2006): 484–99. http://dx.doi.org/10.1198/016214505000001294.

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18

Camerlenghi, Federico, Antonio Lijoi, Peter Orbanz, and Igor Prünster. "Distribution theory for hierarchical processes." Annals of Statistics 47, no. 1 (February 2019): 67–92. http://dx.doi.org/10.1214/17-aos1678.

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19

Thomas, Robert, Marilyn Eichelberger, and Missey Lee. "The Theory of Risk Uncertainty Reduction." Journal of System Safety 54, no. 2 (October 1, 2018): 11–18. http://dx.doi.org/10.56094/jss.v54i2.73.

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The objective of this research is to examine the character of safety programs in not only reducing risk, but also in reducing relative risk uncertainty. This paper approximates the distributions of both the probability and severity of a mishap as lognormal and examines the likely behavior of the co-distribution as the safety process is executed. This paper also shows how differential forces across the risk plane reduce both the risk itself and the relative uncertainty in the risk at the same time. With this new approach, risk now becomes a quantitative item with a known probability distribution, providing a new metric for safety program effectiveness.

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20

Zacks, Shelemyahu, Allan Stuart, and J. Keith Ord. "Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory." Journal of the American Statistical Association 84, no. 407 (September 1989): 841. http://dx.doi.org/10.2307/2289685.

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21

Nakahara, Yuta, Shota Saito, Akira Kamatsuka, and Toshiyasu Matsushima. "Probability Distribution on Full Rooted Trees." Entropy 24, no. 3 (February 24, 2022): 328. http://dx.doi.org/10.3390/e24030328.

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The recursive and hierarchical structure of full rooted trees is applicable to statistical models in various fields, such as data compression, image processing, and machine learning. In most of these cases, the full rooted tree is not a random variable; as such, model selection to avoid overfitting is problematic. One method to solve this problem is to assume a prior distribution on the full rooted trees. This enables the optimal model selection based on Bayes decision theory. For example, by assigning a low prior probability to a complex model, the maximum a posteriori estimator prevents the selection of the complex one. Furthermore, we can average all the models weighted by their posteriors. In this paper, we propose a probability distribution on a set of full rooted trees. Its parametric representation is suitable for calculating the properties of our distribution using recursive functions, such as the mode, expectation, and posterior distribution. Although such distributions have been proposed in previous studies, they are only applicable to specific applications. Therefore, we extract their mathematically essential components and derive new generalized methods to calculate the expectation, posterior distribution, etc.

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22

Eliwa, Mohamed S., Muhammad Ahsan-ul-Haq, Amani Almohaimeed, Afrah Al-Bossly, and Mahmoud El-Morshedy. "Discrete Extension of Poisson Distribution for Overdispersed Count Data: Theory and Applications." Journal of Mathematics 2023 (February 13, 2023): 1–15. http://dx.doi.org/10.1155/2023/2779120.

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In this study, a new one-parameter discrete probability distribution is introduced for overdispersed count data based on a combining approach. The important statistical properties can be expressed in closed forms including factorial moments, moment generating function, dispersion index, coefficient of variation, coefficient of skewness, coefficient of kurtosis, value at risk, and tail value at risk. Moreover, four classical parameter estimation methods have been discussed for this new distribution. A simulation study was conducted to evaluate the performance of different estimators based on the biases, mean related-errors, and mean square errors of the estimators. In the end, real data sets from different fields are analyzed to verify the usefulness of the new probability mass function over some notable discrete distributions. It is manifested that the new discrete probability distribution provides an adequate fit than these distributions.

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23

Balkema, A. A., and L. De Haan. "A convergence rate in extreme-value theory." Journal of Applied Probability 27, no. 3 (September 1990): 577–85. http://dx.doi.org/10.2307/3214542.

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A uniform convergence rate is determined for maxima of i.i.d. random variables from a distribution in the domain of attraction of the double-exponential distribution. The result is proved under a second-order condition on the underlying distribution parallelling the one given in Smith (1982) for the domain of attraction of the bounded-below and bounded-above families of limit distributions.

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24

Balkema, A. A., and L. De Haan. "A convergence rate in extreme-value theory." Journal of Applied Probability 27, no. 03 (September 1990): 577–85. http://dx.doi.org/10.1017/s0021900200039127.

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A uniform convergence rate is determined for maxima of i.i.d. random variables from a distribution in the domain of attraction of the double-exponential distribution. The result is proved under a second-order condition on the underlying distribution parallelling the one given in Smith (1982) for the domain of attraction of the bounded-below and bounded-above families of limit distributions.

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25

Harris, J. A., and B. J. Adams. "Probabilistic assessment of urban runoff erosion potential." Canadian Journal of Civil Engineering 33, no. 3 (March 1, 2006): 307–18. http://dx.doi.org/10.1139/l05-114.

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At the planning or screening level of urban development, analytical modeling using derived probability distribution theory is a viable alternative to continuous simulation, offering considerably less computational effort. A new set of analytical probabilistic models is developed for predicting the erosion potential of urban stormwater runoff. The marginal probability distributions for the duration of a hydrograph in which the critical channel velocity is exceeded (termed exceedance duration) are computed using derived probability distribution theory. Exceedance duration and peak channel velocity are two random variables upon which erosion potential is functionally dependent. Reasonable agreement exists between the derived marginal probability distributions for exceedance duration and continuous EPA Stormwater Management Model (SWMM) simulations at more common return periods. It is these events of lower magnitude and higher frequency that are the most significant to erosion-potential prediction. Key words: erosion, stormwater management, derived probability distribution, exceedance duration.

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26

Bakouch, Hassan S. "Elements of Distribution Theory." Journal of the Royal Statistical Society: Series A (Statistics in Society) 170, no. 4 (October 2007): 1186–87. http://dx.doi.org/10.1111/j.1467-985x.2007.00506_16.x.

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27

Verdú, Sergio. "The Cauchy Distribution in Information Theory." Entropy 25, no. 2 (February 13, 2023): 346. http://dx.doi.org/10.3390/e25020346.

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The Gaussian law reigns supreme in the information theory of analog random variables. This paper showcases a number of information theoretic results which find elegant counterparts for Cauchy distributions. New concepts such as that of equivalent pairs of probability measures and the strength of real-valued random variables are introduced here and shown to be of particular relevance to Cauchy distributions.

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28

Abdul, Abdul, K. Mohana, David Winster P. Devanayan, Krishnaprasad Krishnaprasad, D. Sandhya, and Pradeep Kumar SV. "Generalized p-Transmuted Neutrosophic Distributions: Theory and its Applications." International Journal of Neutrosophic Science 24, no. 3 (2024): 201–19. http://dx.doi.org/10.54216/ijns.240318.

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The study of neutrosophy offers a fresh approach for handling uncertain data with adaptability. This article explores the application of neutrosophic probability distribution in constructing a transmuted neutrosophic framework. Specifically, it introduces a generalized transmuted neutrosophic distribution. Building upon this generalization, quadratic and cubic transmuted distributions are developed and examined alongside certain lifetime distributions serving as foundational neutrosophic models. Additionally, an empirical investigation is conducted to assess the practicality and versatility of these distributions in real-world contexts.

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29

Anderson, Johan, and Eun-jin Kim. "Analytical theory of the probability distribution function of structure formation." Physics of Plasmas 15, no. 8 (August 2008): 082312. http://dx.doi.org/10.1063/1.2973177.

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30

Fahidy, Thomas Z. "Electrochemical horizons for the Poisson-lognormal distribution of probability theory." Journal of Electroanalytical Chemistry 581, no. 1 (July 2005): 11–15. http://dx.doi.org/10.1016/j.jelechem.2005.03.038.

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31

Jodlbauer, Herbert, Matthias Dehmer, and Sonja Strasser. "A hybrid binomial inverse hypergeometric probability distribution: Theory and applications." Applied Mathematics and Computation 338 (December 2018): 44–54. http://dx.doi.org/10.1016/j.amc.2018.05.063.

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32

Dombi, József, Tamás Jónás, Zsuzsanna E. Tóth, and Gábor Árva. "The omega probability distribution and its applications in reliability theory." Quality and Reliability Engineering International 35, no. 2 (November 13, 2018): 600–626. http://dx.doi.org/10.1002/qre.2425.

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33

Ozaki, Vitor Augusto, Ricardo Olinda, Priscila Neves Faria, and Rogério Costa Campos. "Estimation of the agricultural probability of loss: evidence for soybean in Paraná state." Revista de Economia e Sociologia Rural 52, no. 1 (March 2014): 25–40. http://dx.doi.org/10.1590/s0103-20032014000100002.

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In any agricultural insurance program, the accurate quantification of the probability of the loss has great importance. In order to estimate this quantity, it is necessary to assume some parametric probability distribution. The objective of this work is to estimate the probability of loss using the theory of the extreme values modeling the left tail of the distribution. After that, the estimated values will be compared to the values estimated under the normality assumption. Finally, we discuss the implications of assuming a symmetrical distribution instead of a more flexible family of distributions when estimating the probability of loss and pricing the insurance contracts. Results show that, for the selected regions, the probability distributions present a relative degree of skewness. As a consequence, the probability of loss is quite different from those estimated supposing the Normal distribution, commonly used by Brazilian insurers.

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34

Monahan, Adam Hugh. "The Probability Distribution of Sea Surface Wind Speeds. Part I: Theory and SeaWinds Observations." Journal of Climate 19, no. 4 (February 15, 2006): 497–520. http://dx.doi.org/10.1175/jcli3640.1.

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Abstract The probability distribution of sea surface wind speeds, w, is considered. Daily SeaWinds scatterometer observations are used for the characterization of the moments of sea surface winds on a global scale. These observations confirm the results of earlier studies, which found that the two-parameter Weibull distribution provides a good (but not perfect) approximation to the probability density function of w. In particular, the observed and Weibull probability distributions share the feature that the skewness of w is a concave upward function of the ratio of the mean of w to its standard deviation. The skewness of w is positive where the ratio is relatively small (such as over the extratropical Northern Hemisphere), the skewness is close to zero where the ratio is intermediate (such as the Southern Ocean), and the skewness is negative where the ratio is relatively large (such as the equatorward flank of the subtropical highs). An analytic expression for the probability density function of w, derived from a simple stochastic model of the atmospheric boundary layer, is shown to be in good qualitative agreement with the observed relationships between the moments of w. Empirical expressions for the probability distribution of w in terms of the mean and standard deviation of the vector wind are derived using Gram–Charlier expansions of the joint distribution of the sea surface wind vector components. The significance of these distributions for improvements to calculations of averaged air–sea fluxes in diagnostic and modeling studies is discussed.

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Manu, Faruk, U. Usman, and A. Audu. "NEW LIFE-TIME CONTINUOUS PROBABILITY DISTRIBUTION WITH FLEXIBLE FAILURE RATE FUNCTION." FUDMA JOURNAL OF SCIENCES 7, no. 1 (November 29, 2023): 323–29. http://dx.doi.org/10.33003/fjs-2023-0701-2070.

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Recently, the area of distribution theory has been receiving increased interest in generating or defining new classes of continuous probability distributions by way of extending the existing distributions. The new generated distribution is expected to be more flexible and have wider acceptability in modeling and predicting real world data sets. In this research, we proposed and study an extension of Ikum distribution using Zubai G- Family (2018) of distribution called Z-Ikum distribution. Expression of some basic structural properties of the new distribution such as cdf, pdf, quantile functions, moments, moment generating function, characteristics function and order statistics was derived. Survival function, hazard rate function, commutative hazard rate function and reversed hazard rate function was also discussed. Plots of the hazard rate function showincrease, decrease and bathtub shapes.Estimation of the proposed distribution parameters was carried out using MLE method. Performance of the parameter estimation was also evaluated via simulation studies. Result of the simulation studies indicates that our estimator is consistent. Three life data sets were used to evaluate the performance of our proposed distribution over some existing distributions. Result of the empherical study revealed that our proposed distribution performwell in modeling real life data than the competing distributions.

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36

Nayak, Tapan Kumar. "Multivariate Lomax distribution: properties and usefulness in reliability theory." Journal of Applied Probability 24, no. 1 (March 1987): 170–77. http://dx.doi.org/10.2307/3214068.

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A model incorporating the effect of a common environment on several components (structurally independent) of a system is developed. A multivariate generalization of the Lomax (Pareto type 2) distribution is obtained by mixing exponential variables. Its relationship to other multivariate distributions is discussed. Several properties of this distribution are reported and their usefulness in reliability theory indicated. Finally, a further generalization of this multivariate Lomax distribution is presented.

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37

Chicano, Francisco, Andrew M. Sutton, L. Darrell Whitley, and Enrique Alba. "Fitness Probability Distribution of Bit-Flip Mutation." Evolutionary Computation 23, no. 2 (June 2015): 217–48. http://dx.doi.org/10.1162/evco_a_00130.

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Bit-flip mutation is a common mutation operator for evolutionary algorithms applied to optimize functions over binary strings. In this paper, we develop results from the theory of landscapes and Krawtchouk polynomials to exactly compute the probability distribution of fitness values of a binary string undergoing uniform bit-flip mutation. We prove that this probability distribution can be expressed as a polynomial in p, the probability of flipping each bit. We analyze these polynomials and provide closed-form expressions for an easy linear problem (Onemax), and an NP-hard problem, MAX-SAT. We also discuss a connection of the results with runtime analysis.

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38

Bantan, Rashad A. R., Farrukh Jamal, Christophe Chesneau, and Mohammed Elgarhy. "Theory and Applications of the Unit Gamma/Gompertz Distribution." Mathematics 9, no. 16 (August 5, 2021): 1850. http://dx.doi.org/10.3390/math9161850.

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Unit distributions are commonly used in probability and statistics to describe useful quantities with values between 0 and 1, such as proportions, probabilities, and percentages. Some unit distributions are defined in a natural analytical manner, and the others are derived through the transformation of an existing distribution defined in a greater domain. In this article, we introduce the unit gamma/Gompertz distribution, founded on the inverse-exponential scheme and the gamma/Gompertz distribution. The gamma/Gompertz distribution is known to be a very flexible three-parameter lifetime distribution, and we aim to transpose this flexibility to the unit interval. First, we check this aspect with the analytical behavior of the primary functions. It is shown that the probability density function can be increasing, decreasing, “increasing-decreasing” and “decreasing-increasing”, with pliant asymmetric properties. On the other hand, the hazard rate function has monotonically increasing, decreasing, or constant shapes. We complete the theoretical part with some propositions on stochastic ordering, moments, quantiles, and the reliability coefficient. Practically, to estimate the model parameters from unit data, the maximum likelihood method is used. We present some simulation results to evaluate this method. Two applications using real data sets, one on trade shares and the other on flood levels, demonstrate the importance of the new model when compared to other unit models.

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39

Díaz-García, José A. "Distribution theory of quadratic forms for matrix multivariate elliptical distribution." Journal of Statistical Planning and Inference 143, no. 8 (August 2013): 1330–42. http://dx.doi.org/10.1016/j.jspi.2013.03.024.

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40

Kondo, Takefumi. "Probability distribution of metric measure spaces." Differential Geometry and its Applications 22, no. 2 (March 2005): 121–30. http://dx.doi.org/10.1016/j.difgeo.2004.10.001.

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41

Bratischenko, Vladimir. "A Family of Discrete Distributions with Infinite Moments." System Analysis & Mathematical Modeling 4, no. 3 (December 16, 2022): 200–207. http://dx.doi.org/10.17150/2713-1734.2022.4(3).200-207.

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In probability theory, distribution laws of random variables are distinguished, in which the moments diverge to infinity. Such distributions are important for understanding the features of probability distributions; in addition, they find practical application in the study of estimation procedures. Continuous distributions with infinite moments are known and well studied. The paper proposes a family of distributions of discrete random variables that does not have moments starting from a certain order. The distributions are supplemented with a parameter that linearly affects the mathematical expectation. A method for calculating distribution moments is proposed. The probability generating function is determined.

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POPESCU, D., IONELA MIRELA NEAGOE, SUZANA E. CILIEVICI, DIANA R. CONSTANTIN, and V. I. R. NICULESCU. "ANALYSIS OF THE CRYPTOSPORIDIUM SPP GP60 GENE VARIABILITY APPLYING INFORMATION THEORY." Romanian Journal of Biophysics 34, no. 1 (February 27, 2024): 1–12. http://dx.doi.org/10.59277/rjb.2024.1.01.

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In this paper we used statistical methods to understand the genetic information of DNA considered as a statistical system. The alphabet of a DNA sequence is defined by the four nucleotides: adenine, cytosine, guanine, and thymine. The order of nucleotides along the DNA sequences encodes the genetic information. We have analyzed three Cryptosporidium DNA sequences: one DNA sequence isolated and analyzed in our laboratory and two DNA reference sequences from the public database GenBank. Each DNA sequence is considered as a statistical system and is represented by a random variable and an associate probability distribution. The Shannon entropy, Renyi entropy, Onicescu informational energy and square deviation from uniform distribution are used in order to measure the degree of randomness for the three statistical systems. The similarity and difference between the three DNA sequences of the two Cryptosporidium species (Cryptosporidium hominis and Cryptosporidium parvum) were assessed by calculating the statistical distance between the probability distributions associated with each pair of DNA sequences. Each of the three DNA sequences pairs with one of the other two sequences and forms three pairs of sequences. Using the associated probability distributions, the statistical distance between them can be calculated. Bhattacharyya distance measures similarity degree between the two probability distributions. The Kullback-Leiber and the resistor-average distances measure the difference between the two distributions.

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Haugh, Larry D., N. Balakrishnan, and Asit P. Basu. "The Exponential Distribution: Theory, Methods, and Applications." Journal of the American Statistical Association 92, no. 439 (September 1997): 1221. http://dx.doi.org/10.2307/2965598.

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Wootters, William. "A Classical Interpretation of the Scrooge Distribution." Entropy 20, no. 8 (August 20, 2018): 619. http://dx.doi.org/10.3390/e20080619.

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The Scrooge distribution is a probability distribution over the set of pure states of a quantum system. Specifically, it is the distribution that, upon measurement, gives up the least information about the identity of the pure state compared with all other distributions that have the same density matrix. The Scrooge distribution has normally been regarded as a purely quantum mechanical concept with no natural classical interpretation. In this paper, we offer a classical interpretation of the Scrooge distribution viewed as a probability distribution over the probability simplex. We begin by considering a real-amplitude version of the Scrooge distribution for which we find that there is a non-trivial but natural classical interpretation. The transition to the complex-amplitude case requires a step that is not particularly natural but that may shed light on the relation between quantum mechanics and classical probability theory.

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Abdullahi, Usman Aliyu, Ahmad Abubakar Suleiman, Aliyu Ismail Ishaq, Abubakar Usman, and Aminu Suleiman. "The Maxwell – Exponential Distribution: Theory and Application to Lifetime Data." Journal of Statistical Modelling and Analytics 3, no. 2 (October 15, 2021): 65–80. http://dx.doi.org/10.22452/josma.vol3no2.4.

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Two parameters Maxwell – Exponential distribution was proposed using the Maxwell generalized family of distribution. The probability density function, cumulative distribution function, survival function, hazard function, quantile function, and statistical properties of the proposed distribution are discussed. The parameters of the proposed distribution have been estimated using the maximum likelihood estimation method. The potentiality of the estimators was shown using a simulation study. The overall assessment of the performance of Maxwell - Exponential distribution was determined by using two real-life datasets. Our findings reveal that the Maxwell – Exponential distribution is more flexible compared to other competing distributions as it has the least value of information criteria.

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Tarasov, Vasily E. "General Nonlocal Probability of Arbitrary Order." Entropy 25, no. 6 (June 10, 2023): 919. http://dx.doi.org/10.3390/e25060919.

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Using the Luchko’s general fractional calculus (GFC) and its extension in the form of the multi-kernel general fractional calculus of arbitrary order (GFC of AO), a nonlocal generalization of probability is suggested. The nonlocal and general fractional (CF) extensions of probability density functions (PDFs), cumulative distribution functions (CDFs) and probability are defined and its properties are described. Examples of general nonlocal probability distributions of AO are considered. An application of the multi-kernel GFC allows us to consider a wider class of operator kernels and a wider class of nonlocality in the probability theory.

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Harada, Ryo. "Operational and Harmonic-Analytic Aspects of Quasi-Probability Distributions." Open Systems & Information Dynamics 18, no. 03 (September 2011): 301–20. http://dx.doi.org/10.1142/s1230161211000200.

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Husimi distributions [11] and Wigner distributions [20] are well-known quasi-probability distributions which appear in several contexts. In this paper, we show some remarkable aspects of these distribution functions related to geometric structures of generalized coherent state systems [15,16] and operational quantum physics [7], and a scheme of formulating generalized version of quasi-probability distributions. Our scheme gives concrete formulae of quasi-probability distributions in more direct way from the theory of coherent state systems and clarify their operational meanings, especially of Husimi distributions and mutual relation between Husimi distributions and other classes of quasi-probability distributions.

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Kumar, P. "Probability distributions conditioned by the available information: Gamma distribution and moments." Computers & Mathematics with Applications 52, no. 3-4 (August 2006): 289–304. http://dx.doi.org/10.1016/j.camwa.2006.08.020.

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Chachi, Jalal. "On Distribution Characteristics of a Fuzzy Random Variable." Austrian Journal of Statistics 47, no. 2 (February 2, 2018): 53–67. http://dx.doi.org/10.17713/ajs.v47i2.581.

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In this paper, rst a new notion of fuzzy random variables is introduced. Then, usingclassical techniques in Probability Theory, some aspects and results associated to a randomvariable (including expectation, variance, covariance, correlation coecient, etc.) will beextended to this new environment. Furthermore, within this framework, we can use thetools of general Probability Theory to dene fuzzy cumulative distribution function of afuzzy random variable.

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Heydari, Tahereh, Karim Zare, Soheil Shokri, Zahra Khodadadi, and Zahra Almaspoor. "A New Sine-Based Probabilistic Approach: Theory and Monte Carlo Simulation with Reliability Application." Journal of Mathematics 2024 (January 22, 2024): 1–19. http://dx.doi.org/10.1155/2024/9593193.

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Data modeling is a very crucial stage for decision making in applied sectors. Probability distributions are considered important tools for decision making. So far, numerous probability distributions have been developed and implemented. Most of these distributions are developed by introducing from one to eight additional parameters. Sometimes, the addition of new parameters leads to re-parameterization problems. To avoid such issues, we introduce a novel probabilistic approach. The proposed approach may be termed as a new weighted sine-G method. The beauty and key advantage of the new weighted sine-G method are that it has no additional parameters. Through using the new weighted sine-G method, a new weighted sine-Weibull distribution is introduced, which is a modification of the Weibull distribution. The estimators of the new model are also derived. Furthermore, a simulation study is carried out to evaluate the estimators of the new weighted sine-Weibull distribution. Finally, a practical application from the reliability sector is considered to evaluate the new weighted sine-Weibull distribution. Based on certain decision tools, it is observed that the proposed model is the best competing distribution for applying it in the reliability sector.

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