scholarly journals On a new distribution based on the arccosine function

2021 ◽  
Author(s):  
Christophe Chesneau ◽  
Lishamol Tomy ◽  
Jiju Gillariose
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Hazard Rate ◽  
Survival Function ◽  
Real Data ◽  
Cumulative Distribution ◽  
Data Sets ◽  
Power Functions ◽  
New Distribution

AbstractThis note focuses on a new one-parameter unit probability distribution centered around the inverse cosine and power functions. A special case of this distribution has the exact inverse cosine function as a probability density function. To our knowledge, despite obvious mathematical interest, such a probability density function has never been considered in Probability and Statistics. Here, we fill this gap by pointing out the main properties of the proposed distribution, from both the theoretical and practical aspects. Specifically, we provide the analytical form expressions for its cumulative distribution function, survival function, hazard rate function, raw moments and incomplete moments. The asymptotes and shape properties of the probability density and hazard rate functions are described, as well as the skewness and kurtosis properties, revealing the flexible nature of the new distribution. In particular, it appears to be “round mesokurtic” and “left skewed”. With these features in mind, special attention is given to find empirical applications of the new distribution to real data sets. Accordingly, the proposed distribution is compared with the well-known power distribution by means of two real data sets.

scholarly journals Alpha Power Transformed Lindley Distribution

2021 ◽  
pp. 130-142
Author(s):  
L A Rosa ◽  
S Nurrohmah ◽  
I Fithriani
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Hazard Rate ◽  
Survival Function ◽  
Cumulative Distribution ◽  
Likelihood Method ◽  
Alpha Power ◽  
Lindley Distribution ◽  
Modelling Data

The one parameter Lindley distribustion (theta) has been widely used in various field such as biology, technique, medical, and industries. Lindley distribution is capable for modelling data with monotone increasing hazard rate. However, in real life, there are situations where the hazard rate is not monotone. Therefore, to enhance the Lindley distribution capabilitiesfor modelling data, a modification can be used by using Alpha Power Transformed method. The result of the modification of Lindley distribution is commonly called Alpha Power Transformed Lindley distribution (APTL) distribution that has two parameters (alpha, theta). This new APTL distribution is appropriate in modelling data with decreasing or unimodal shaped of probability density function, and has hazard rates with increasing, decreasing, and upside-down bathtub shaped. The properties of the proposed distribution are discussed include probability density function, cumulative distribution function, survival function, hazard rate function, moment generating function, and rth moment. Themodel parameters are obtained using maximum likelihood method. The waiting time data is used as an illustration to describe the utility of APTL distribution.

scholarly journals A Probability Density Function Generator Based on Deep Learning

2018 ◽  
Author(s):  
Chi-Hua Chen ◽  
Fangying Song ◽  
Feng-Jang Hwang ◽  
Ling Wu
Keyword(s):  
Deep Learning ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Probability Distributions ◽  
Real Data ◽  
Learning Model ◽  
Cumulative Distribution ◽  
Function Generator ◽  
Deep Learning Model

To generate a probability density function (PDF) for fitting probability distributions of real data, this study proposes a deep learning method which consists of two stages: (1) a training stage for estimating the cumulative distribution function (CDF) and (2) a performing stage for predicting the corresponding PDF. The CDFs of common probability distributions can be adopted as activation functions in the hidden layers of the proposed deep learning model for learning actual cumulative probabilities, and the differential equation of trained deep learning model can be used to estimate the PDF. To evaluate the proposed method, numerical experiments with single and mixed distributions are performed. The experimental results show that the values of both CDF and PDF can be precisely estimated by the proposed method.

Fourier Transforms in Probability, Random Variables and Stochastic Processes

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Fourier Analysis ◽  
Stochastic Processes ◽  
Probability Density ◽  
Density Function ◽  
Fourier Transforms ◽  
Random Variables ◽  
Cumulative Distribution ◽  
The Face

In this Chapter, we present application of Fourier analysis to probability, random variables and stochastic processes [1089, 1097, 1387, 1329]. Arandom variable, X, is the assignment of a number to the outcome of a random experiment. We can, for example, flip a coin and assign an outcome of a heads as X = 1 and a tails X = 0. Often the number is equated to the numerical outcome of the experiment, such as the number of dots on the face of a rolled die or the measurement of a voltage in a noisy circuit. The cumulative distribution function is defined by FX(x) = Pr[X ≤ x]. (4.1) The probability density function is the derivative fX(x) = d /dxFX(x). Our treatment of random variables focuses on use of Fourier analysis. Due to this viewpoint, the development we use is unconventional and begins immediately in the next section with discussion of properties of the probability density function.

Event Location with Sparse Data: When Probabilistic Global Search is Important

2020 ◽  
Author(s):  
Stephen Arrowsmith ◽  
Junghyun Park ◽  
Il-Young Che ◽  
Brian Stump ◽  
Gil Averbuch
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Arrival Time ◽  
Real Data ◽  
Parameter Estimates ◽  
Model Parameter ◽  
Event Location ◽  
Seismic Location ◽  
Probability Density Function Pdf

Abstract Locating events with sparse observations is a challenge for which conventional seismic location techniques are not well suited. In particular, Geiger’s method and its variants do not properly capture the full uncertainty in model parameter estimates, which is characterized by the probability density function (PDF). For sparse observations, we show that this PDF can deviate significantly from the ellipsoidal form assumed in conventional methods. Furthermore, we show how combining arrival time and direction-of-arrival constraints—as can be measured by three-component polarization or array methods—can significantly improve the precision, and in some cases reduce bias, in location solutions. This article explores these issues using various types of synthetic and real data (including single-component seismic, three-component seismic, and infrasound).

scholarly journals Bivariate Burr Type III Distribution: Estimation and Prediction

2021 ◽  
pp. 16-36
Author(s):  
A. A. M. Mahmoud ◽  
R. M. Refaey ◽  
G. R. AL-Dayian ◽  
A. A. EL-Helbawy
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Real Life ◽  
Bivariate Distribution ◽  
Cumulative Distribution ◽  
Data Set ◽  
Squared Error Loss Function ◽  
Type Iii ◽  
Estimation And Prediction

In this paper, a bivariate Burr Type III distribution is constructed and some of its statistical properties such as bivariate probability density function and its marginal, joint cumulative distribution and its marginal, reliability and hazard rate functions are studied. The joint probability density function and the joint cumulative distribution are given in closed forms. The joint expectation of this distribution is proposed. The maximum likelihood estimation and prediction for a future observation are derived. Also, Bayesian estimation and prediction are considered under squared error loss function. The performance of the proposed bivariate distribution is examined using a simulation study. Finally, a data set is analyzed under the proposed distribution to illustrate its flexibility for real-life application.

The joint asymptotic distribution of the k-smallest sample spacings

1969 ◽  
Vol 6 (02) ◽  
pp. 442-448
Author(s):  
Lionel Weiss
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Cumulative Distribution Function ◽  
Open Interval ◽  
Cumulative Distribution ◽  
The Right ◽  
Sample Spacings ◽  
Independent Identically Distributed

Suppose Q 1 ⋆, … Q n ⋆ are independent, identically distributed random variables, each with probability density function f(x), cumulative distribution function F(x), where F(1) – F(0) = 1, f(x) is continuous in the open interval (0, 1) and continuous on the right at x = 0 and on the left at x = 1, and there exists a positive C such that f(x) > C for all x in (0, l). f(0) is defined as f(0+), f(1) is defined as f(1–).

Empirical Analyses of Income: Finland (2009) and Australia (1967-1968)

2021 ◽  
pp. 35-53
Author(s):  
Johan Fellman
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Lorenz Curve ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Income Data ◽  
Difference Quotients ◽  
The Lorenz Curve

Analyses of income data are often based on assumptions concerning theoretical distributions. In this study, we apply statistical analyses, but ignore specific distribution models. The main income data sets considered in this study are taxable income in Finland (2009) and household income in Australia (1967-1968). Our intention is to compare statistical analyses performed without assumptions of the theoretical models with earlier results based on specific models. We have presented the central objects, probability density function, cumulative distribution function, the Lorenz curve, the derivative of the Lorenz curve, the Gini index and the Pietra index. The trapezium rule, Simpson´s rule, the regression model and the difference quotients yield comparable results for the Finnish data, but for the Australian data the differences are marked. For the Australian data, the discrepancies are caused by limited data. JEL classification numbers: D31, D63, E64. Keywords: Cumulative distribution function, Probability density function, Mean, quantiles, Lorenz curve, Gini coefficient, Pietra index, Robin Hood index, Trapezium rule, Simpson´s rule, Regression models, Difference quotients, Derivative of Lorenz curve

scholarly journals A NOTE CONCERNING THE DISTANCES OF UNIFORMLY DISTRIBUTED POINTS FROM THE CENTRE OF A RECTANGLE

2012 ◽  
Vol 87 (1) ◽  
pp. 115-119 ◽  
Author(s):  
ROBERT STEWART ◽  
HONG ZHANG
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Density Function ◽  
Cumulative Distribution Function ◽  
Average Distance ◽  
Cumulative Distribution

AbstractGiven a rectangle containing uniformly distributed random points, how far are the points from the rectangle’s centre? In this paper we provide closed-form expressions for the cumulative distribution function and probability density function that characterise the distance. An expression for the average distance to the centre of the rectangle is also provided.

scholarly journals Qualitative Properties of Randomized Maximum Entropy Estimates of Probability Density Functions

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 548
Author(s):  
Yuri S. Popkov
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Maximum Entropy ◽  
Density Function ◽  
Lagrange Multipliers ◽  
Real Data ◽  
Model Parameters ◽  
Lake Area ◽  
Data Set ◽  
Maximum Entropy Estimation

The problem of randomized maximum entropy estimation for the probability density function of random model parameters with real data and measurement noises was formulated. This estimation procedure maximizes an information entropy functional on a set of integral equalities depending on the real data set. The technique of the Gâteaux derivatives is developed to solve this problem in analytical form. The probability density function estimates depend on Lagrange multipliers, which are obtained by balancing the model’s output with real data. A global theorem for the implicit dependence of these Lagrange multipliers on the data sample’s length is established using the rotation of homotopic vector fields. A theorem for the asymptotic efficiency of randomized maximum entropy estimate in terms of stationary Lagrange multipliers is formulated and proved. The proposed method is illustrated on the problem of forecasting of the evolution of the thermokarst lake area in Western Siberia.

scholarly journals Modified Weighted Pareto Distribution Type I (MWPDTI)

2020 ◽  
Vol 17 (3) ◽  
pp. 0869
Author(s):  
Mahmood A. Sahmran
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Pareto Distribution ◽  
Random Variable ◽  
Reliability Function ◽  
Cumulative Distribution ◽  
Type I ◽  
Standard Distribution ◽  
Distribution Type

In this paper, the Azzallini’s method used to find a weighted distribution derived from the standard Pareto distribution of type I (SPDTI) by inserting the shape parameter (θ) resulting from the above method to cover the period (0, 1] which was neglected by the standard distribution. Thus, the proposed distribution is a modification to the Pareto distribution of the first type, where the probability of the random variable lies within the period  The properties of the modified weighted Pareto distribution of the type I (MWPDTI) as the probability density function ,cumulative distribution function, Reliability function , Moment and  the hazard function are found. The behaviour of probability density function for MWPDTI distribution by representing the values of    This means, the probability density function of this distribution treats the period (0,1] which is ignore in SPDTI.

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