Probability Density Functions

2019 ◽  
pp. 108-130
Author(s):  
Therese M. Donovan ◽  
Ruth M. Mickey
Keyword(s):  
Bayesian Inference ◽  
Probability Distribution ◽  
Probability Density ◽  
Probability Distributions ◽  
Random Variable ◽  
Probability Density Functions ◽  
Density Functions ◽  
Normal Distributions ◽  
Likelihood Profile ◽  
Variable Probability

This chapter builds on probability distributions. Its focus is on general concepts associated with probability density functions (pdf’s), which are distributions associated with continuous random variables. The continuous uniform and normal distributions are highlighted as examples of pdf’s. These and other pdf’s can be used to specify prior distributions, likelihoods, and/or posterior distributions in Bayesian inference. Although this chapter specifically focuses on the continuous uniform and normal distributions, the concepts discussed in this chapter will apply to other continuous probability distributions. By the end of the chapter, the reader should be able to define and use the following terms for a continuous random variable: random variable, probability distribution, parameter, probability density, likelihood, and likelihood profile.

scholarly journals Characterizations of Some Probability Distributions with Completely Monotonic Density Functions

2021 ◽  
pp. 51-64
Author(s):  
Mohammad Shakil ◽  
Dr. Mohammad Ahsanullah ◽  
Dr. B. M. G. Kibria Kibria
Keyword(s):  
Probability Distribution ◽  
Probability Density ◽  
Probability Distributions ◽  
Random Variable ◽  
Record Values ◽  
Density Functions ◽  
Exponential Distributions ◽  
Completely Monotonic ◽  
Percentage Points ◽  
Generalized Gamma Function

For a non-negative continuous random variable , Chaudhry and Zubair (2002, p. 19) introduced a probability distribution with a completely monotonic probability density function based on the generalized gamma function, and called it the Macdonald probability function. In this paper, we establish various basic distributional properties of Chaudhry and Zubair’s Macdonald probability distribution. Since the percentage points of a given distribution are important for any statistical applications, we have also computed the percentage points for different values of the parameter involved. Based on these properties, we establish some new characterization results of Chaudhry and Zubair’s Macdonald probability distribution by the left and right truncated moments, order statistics and record values. Characterizations of certain other continuous probability distributions with completely monotonic probability density functions such as Mckay, Pareto and exponential distributions are also discussed by the proposed characterization techniques.   

A New Approach to Probability in Engineering Design and Optimization

1984 ◽  
Vol 106 (1) ◽  
pp. 5-10 ◽  
Author(s):  
J. N. Siddall
Keyword(s):  
Probability Density ◽  
Engineering Design ◽  
Probability Distributions ◽  
Probability Density Functions ◽  
Density Functions ◽  
New Approach ◽  
Design And Optimization ◽  
Evolutionary Technique ◽  
Probability And Statistics

The anomalous position of probability and statistics in both mathematics and engineering is discussed, showing that there is little consensus on concepts and methods. For application in engineering design, probability is defined as strictly subjective in nature. It is argued that the use of classical methods of statistics to generate probability density functions by estimating parameters for assumed theoretical distributions should be used with caution, and that the use of confidence limits is not really meaningful in a design context. Preferred methods are described, and a new evolutionary technique for developing probability distributions of new random variables is proposed. Although Bayesian methods are commonly considered to be subjective, it is argued that, in the engineering sense, they are really not. A general formulation of the probabilistic optimization problem is described, including the role of subjective probability density functions.

scholarly journals Computing the two first probability density functions of the random Cauchy-Euler differential equation: Study about regular-singular points

2017 ◽  
Vol 2 (1) ◽  
pp. 213-224 ◽  
Author(s):  
J.-C. Cortés ◽  
A. Navarro-Quiles ◽  
J.-V. Romero ◽  
M.-D. Roselló
Keyword(s):  
Differential Equation ◽  
Probability Density ◽  
Random Variable ◽  
Probability Density Functions ◽  
Point Of View ◽  
Density Functions ◽  
Statistical Point ◽  
Covariance Functions ◽  
Infinite Point ◽  
Homogeneous Linear

AbstractIn this paper the randomized Cauchy-Euler differential equation is studied. With this aim, from a statistical point of view, both the first and second probability density functions of the solution stochastic process are computed. Then, the main statistical functions, namely, the mean, the variance and the covariance functions are determined as well. The study includes the computation of the first and second probability density functions of the regular-singular infinite point via an adequate mapping transforming the problem about the origin. The study is strongly based upon the Random Variable Transformation technique along with some results that have been recently published by some of authors to the random homogeneous linear second-order differential equation. Finally, an illustrative example is shown.

scholarly journals Exact and Approximate Weighted Model Integration with Probability Density Functions Using Knowledge Compilation

2019 ◽  
Vol 33 ◽  
pp. 7825-7833 ◽  
Author(s):  
Pedro Zuidberg Dos Martires ◽  
Anton Dries ◽  
Luc De Raedt
Keyword(s):  
Probability Density ◽  
Probabilistic Reasoning ◽  
Probability Distributions ◽  
Probability Density Functions ◽  
Knowledge Compilation ◽  
Model Integration ◽  
Weight Functions ◽  
Density Functions ◽  
Model Counting ◽  
Weighted Model

Weighted model counting has recently been extended to weighted model integration, which can be used to solve hybrid probabilistic reasoning problems. Such problems involve both discrete and continuous probability distributions. We show how standard knowledge compilation techniques (to SDDs and d-DNNFs) apply to weighted model integration, and use it in two novel solvers, one exact and one approximate solver. Furthermore, we extend the class of employable weight functions to actual probability density functions instead of mere polynomial weight functions.

Reformative translation model for probability density functions of amplitude processes of crosswind-excited super-tall buildings

2019 ◽  
Vol 23 (5) ◽  
pp. 847-856
Author(s):  
Wei Hao ◽  
Qingshan Yang
Keyword(s):  
Probability Distribution ◽  
Probability Density ◽  
Tall Buildings ◽  
Extreme Value ◽  
Probability Density Functions ◽  
Cumulative Distribution ◽  
Extreme Value Analysis ◽  
Density Functions ◽  
Translation Model ◽  
Non Gaussian

At the vicinity of vortex lock-in wind speed, the nonlinear aerodynamic damping effect of super-tall buildings is significant, which can greatly promote the surge of vortex-induced vibration in the crosswind direction, where the crosswind response characterized by harmonic amplitude shows narrow-band hardening non-Gaussian characteristic with the kurtosis well below 3, and the corresponding probability distribution of amplitude process distinctly differs from that of typical random buffeting response. Although the moment-based Hermite translation model has been widely used for estimating the extreme value distribution of non-Gaussian process, it fails to represent the probability distribution of hardening non-Gaussian amplitude process, notably for the response with a kurtosis close to 1.5. In this study, a new translation model based on orthogonal expansion of random processes is developed for obtaining the non-Gaussian amplitude process from an underlying Gaussian amplitude process, and the probability density function of the non-Gaussian amplitude process is derived by mapping the cumulative distribution function. The coefficients of translation model are determined by minimizing the errors between the estimated probability density functions and target values through nonlinear optimization, and the closed-form semi-empirical formulations, which connect the model coefficients with response kurtosis, are also proposed using least-square curve fitting. Moreover, the effectiveness and monotonicity of the proposed translation model are examined. This model can be readily incorporated into the extreme value analysis of crosswind response and facilitate the evaluation of wind-induced fatigue of super-tall buildings.

The Estimation of Wind Speed and Wind Power Characteristics in Taiwan

2014 ◽  
Vol 535 ◽  
pp. 145-148
Author(s):  
Jeeng Min Ling ◽  
Kunkerati Lublertlop
Keyword(s):  
Wind Speed ◽  
Probability Density ◽  
Wind Power ◽  
Probability Distributions ◽  
Weibull Model ◽  
Probability Density Functions ◽  
Density Functions ◽  
Power Characteristics ◽  
Wind Characteristic ◽  
Better Than

In this paper, the Weibull, Gamma, Lognormal, Rayleigh probability density functions (PDF) were used to statistically analyze the characteristics of wind speed and evaluate the energy based on hourly records from years of 2004 to 2009 at 24 locations in Taiwan. Weibull model shows the best goodness probability density function for estimating behavior of wind characteristic within six years at 7 sites of weather station better than using the Gamma and Rayleigh model. The annual mean wind power density is estimated and compared by different index. The feasibility of probability distributions at different locations were investigated.

scholarly journals Stochastic Predictions of Ore Production in an Underground Limestone Mine Using Different Probability Density Functions: A Comparative Study Using Big Data from ICT System

2021 ◽  
Vol 11 (9) ◽  
pp. 4301
Author(s):  
Dahee Jung ◽  
Jieun Baek ◽  
Yosoon Choi
Keyword(s):  
Big Data ◽  
Probability Distribution ◽  
Probability Density ◽  
Discrete Event Simulation ◽  
Discrete Event ◽  
Real Data ◽  
Probability Density Functions ◽  
Density Functions ◽  
Travel Times ◽  
Event Simulation

This study stochastically predicted ore production through discrete event simulation using four different probability density functions for truck travel times. An underground limestone mine was selected as the study area. The truck travel time was measured by analyzing the big data acquired from information and communications technology (ICT) systems in October 2018, and probability density functions (uniform, triangular, normal, and observed probability distribution of real data) were determined using statistical values. A discrete event simulation model for a truck haulage system was designed, and truck travel times were randomly generated using a Monte Carlo simulation. The ore production that stochastically predicted fifty times for each probability density function was analyzed and represented as a value of lower 10% (P10), 50% (P50), and 90% (P90). Ore production was underestimated when a uniform and triangular distribution was used, as the actual ore production was similar to that of P90. Conversely, the predicted ore production of P50 was relatively consistent with the actual ore production when using the normal and observed probability distribution of real data. The root mean squared error (RMSE) for predicting ore production for ten days in October 2018 was the lowest (24.9 ton/day) when using the observed probability distribution.

Review Of Some Mathematical And Physical Subjects

2006 ◽  
Author(s):  
Abraham Nitzan
Keyword(s):  
Probability Distributions ◽  
Random Variables ◽  
Random Variable ◽  
Probability Density Functions ◽  
Density Functions ◽  
Discrete Random Variables ◽  
Actual Learning ◽  
The Given ◽  
Selection Of ◽  
Comprehensive Discussion

This chapter reviews some subjects in mathematics and physics that are used in different contexts throughout this book. The selection of subjects and the level of their coverage reflect the author’s perception of what potential users of this text were exposed to in their earlier studies. Therefore, only brief overview is given of some subjects while somewhat more comprehensive discussion is given of others. In neither case can the coverage provided substitute for the actual learning of these subjects that are covered in detail by many textbooks. A random variable is an observable whose repeated determination yields a series of numerical values (“realizations” of the random variable) that vary from trial to trial in a way characteristic of the observable. The outcomes of tossing a coin or throwing a die are familiar examples of discrete random variables. The position of a dust particle in air and the lifetime of a light bulb are continuous random variables. Discrete random variables are characterized by probability distributions; Pn denotes the probability that a realization of the given random variable is n. Continuous random variables are associated with probability density functions P(x): P(x1)dx denotes the probability that the realization of the variable x will be in the interval x1 . . . x1+dx.

scholarly journals RSSI Probability Density Functions Comparison Using Jensen-Shannon Divergence and Pearson Distribution

Technologies ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 26
Author(s):  
Antonios Lionis ◽  
Konstantinos P. Peppas ◽  
Hector E. Nistazakis ◽  
Andreas Tsigopoulos
Keyword(s):  
Probability Density ◽  
Atmospheric Turbulence ◽  
Probability Distributions ◽  
Probability Density Functions ◽  
Index Structure ◽  
Optical Channel ◽  
Density Functions ◽  
Free Space Optical ◽  
Leibler Divergence ◽  
Jensen Shannon Divergence

The performance of a free-space optical (FSO) communications link suffers from the deleterious effects of weather conditions and atmospheric turbulence. In order to better estimate the reliability and availability of an FSO link, a suitable distribution needs to be employed. The accuracy of this model depends strongly on the atmospheric turbulence strength which causes the scintillation effect. To this end, a variety of probability density functions were utilized to model the optical channel according to the strength of the refractive index structure parameter. Although many theoretical models have shown satisfactory performance, in reality they can significantly differ. This work employs an information theoretic method, namely the so-called Jensen–Shannon divergence, a symmetrization of the Kullback–Leibler divergence, to measure the similarity between different probability distributions. In doing so, a large experimental dataset of received signal strength measurements from a real FSO link is utilized. Additionally, the Pearson family of continuous probability distributions is also employed to determine the best fit according to the mean, standard deviation, skewness and kurtosis of the modeled data.

Sign in / Sign up

Export Citation Format

Share Document