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.

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 Moments of order statistics from nonidentically Distributed Lomax, exponential Lomax and exponential Pareto Variables

2018 ◽  
Vol 14 (1) ◽  
pp. 7431-7438
Author(s):  
Nasr Ibrahim Rashwan
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Probability Density ◽  
Order Statistics ◽  
Density Function ◽  
Order Statistic ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Numerical Examples ◽  
Moments Of Order Statistics

In this paper, the probability density function and the cumulative distribution function of the rth order statistic arising from independent nonidentically distributed (INID) Lomax, exponential Lomax and exponential Pareto variables are presented. The moments of order statistics from INID Lomax, exponential lomax and exponential Pareto were derived using the technique established by Barakat and Abdelkader. Also, numerical examples are given.

scholarly journals Evolution of bioacoustic correlometers: from setups for analysis of probability density function (PDF), cumulative distribution function (CDF), [spectral] entropy of signal (SE) & quality of masking noise (QMN) to palmtop-like pocket devices

2019 ◽  
Author(s):  
Oleg Gradov ◽  
Eugene Adamovich ◽  
Serge Pankratov
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Spectral Entropy ◽  
Masking Noise ◽  
Probability Density Function Pdf

Evolution of bioacoustic correlometers: from setups for analysis of probability density function (PDF), cumulative distribution function (CDF), [spectral] entropy of signal (SE) & quality of masking noise (QMN) to palmtop-like pocket devices Novel references:

The joint asymptotic distribution of the k-smallest sample spacings

1969 ◽  
Vol 6 (2) ◽  
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 Q1⋆, … Qn⋆ 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–).

scholarly journals Some Generalized Inequalities Involving Local Fractional Integrals and their Applications for Random Variables and Numerical Integration

2016 ◽  
Vol 12 (2) ◽  
pp. 49-65 ◽  
Author(s):  
S. Erden ◽  
M. Z. Sarikaya ◽  
N. Çelik
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Probability Density ◽  
Numerical Integration ◽  
Density Function ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Fractional Integrals ◽  
Grüss Inequality

Abstract We establish generalized pre-Grüss inequality for local fractional integrals. Then, we obtain some inequalities involving generalized expectation, p−moment, variance and cumulative distribution function of random variable whose probability density function is bounded. Finally, some applications for generalized Ostrowski-Grüss inequality in numerical integration are given.

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.

Multivariate probabilistic rock physics models using Kumaraswamy distributions

Geophysics ◽  
2021 ◽  
pp. 1-43
Author(s):  
Dario Grana
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Random Variables ◽  
Rock Physics ◽  
Petrophysical Properties ◽  
Kumaraswamy Distribution ◽  
Model Predictions ◽  
Physics Model ◽  
Rock Physics Model

Rock physics models are physical equations that map petrophysical properties into geophysical variables, such as elastic properties and density. These equations are generally used in quantitative log and seismic interpretation to estimate the properties of interest from measured well logs and seismic data. Such models are generally calibrated using core samples and well log data and result in accurate predictions of the unknown properties. Because the input data are often affected by measurement errors, the model predictions are often uncertain. Instead of applying rock physics models to deterministic measurements, I propose to apply the models to the probability density function of the measurements. This approach has been previously adopted in literature using Gaussian distributions, but for petrophysical properties of porous rocks, such as volumetric fractions of solid and fluid components, the standard probabilistic formulation based on Gaussian assumptions is not applicable due to the bounded nature of the properties, the multimodality, and the non-symmetric behavior. The proposed approach is based on the Kumaraswamy probability density function for continuous random variables, which allows modeling double bounded non-symmetric distributions and is analytically tractable, unlike the Beta or Dirichtlet distributions. I present a probabilistic rock physics model applied to double bounded continuous random variables distributed according to a Kumaraswamy distribution and derive the analytical solution of the posterior distribution of the rock physics model predictions. The method is illustrated for three rock physics models: Raymer’s equation, Dvorkin’s stiff sand model, and Kuster-Toksoz inclusion model.

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