scholarly journals Auto-Transformations of the Probability Density Functions

2021 ◽  
pp. 167-178
Author(s):  
Alexander Harin
Keyword(s):  
Probability Density ◽  
Present Article ◽  
Random Variables ◽  
Probability Density Functions ◽  
Density Functions ◽  
Behavioral Sciences ◽  
Basic Features ◽  
Finite Domains

The two goals of the present article are: 1) To define transformations (named here as auto-transformations) of the probability density functions (PDFs) of random variables into some similar functions having smaller sizes of their domains. 2) To research and outline basic features of these auto-transformations of PDFs. Particularly, auto-transformations from infinite to finite domains are analyzed. The goals are caused by the well-known problems of behavioral sciences.

Classes of probability density functions having Laplace transforms with negative zeros and poles

1987 ◽  
Vol 19 (3) ◽  
pp. 632-651 ◽  
Author(s):  
Ushio Sumita ◽  
Yasushi Masuda
Keyword(s):  
Probability Density ◽  
Practical Importance ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Laplace Transforms ◽  
Probability Density Functions ◽  
Independent Random Variables ◽  
Density Functions ◽  
Exponential Random Variables ◽  
Zeros And Poles

We consider a class of functions on [0,∞), denoted by Ω, having Laplace transforms with only negative zeros and poles. Of special interest is the class Ω+ of probability density functions in Ω. Simple and useful conditions are given for necessity and sufficiency of f ∊ Ω to be in Ω+. The class Ω+ contains many classes of great importance such as mixtures of n independent exponential random variables (CMn), sums of n independent exponential random variables (PF∗n), sums of two independent random variables, one in CMr and the other in PF∗1 (CMPFn with n = r + l) and sums of independent random variables in CMn(SCM). Characterization theorems for these classes are given in terms of zeros and poles of Laplace transforms. The prevalence of these classes in applied probability models of practical importance is demonstrated. In particular, sufficient conditions are given for complete monotonicity and unimodality of modified renewal densities.

Probability and random variables

2019 ◽  
pp. 38-59
Author(s):  
M. D. Edge
Keyword(s):  
Probability Density ◽  
Random Variables ◽  
Bayes Theorem ◽  
Probability Density Functions ◽  
Density Functions ◽  
Conditional Probabilities ◽  
Discrete Random Variables ◽  
New Information ◽  
Probability Mass ◽  
Mass Functions

This chapter considers the rules of probability. Probabilities are non-negative, they sum to one, and the probability that either of two mutually exclusive events occurs is the sum of the probability of the two events. Two events are said to be independent if the probability that they both occur is the product of the probabilities that each event occurs. Bayes’ theorem is used to update probabilities on the basis of new information, and it is shown that the conditional probabilities P(A|B) and P(B|A) are not the same. Finally, the chapter discusses ways in which distributions of random variables can be described, using probability mass functions for discrete random variables and probability density functions for continuous random variables.

scholarly journals Stochastic Simulation of Production Processes – Selected Issues

2021 ◽  
Vol 1 (1) ◽  
Author(s):  
Ryszard SNOPKOWSKI ◽  
Marta SUKIENNIK ◽  
Aneta NAPIERAJ
Keyword(s):  
Production Process ◽  
Probability Density ◽  
Stochastic Simulation ◽  
Random Variables ◽  
Probability Density Functions ◽  
Density Functions ◽  
Production Processes

The article presents selected issues in the field of stochastic simulation of production process-es. Attention was drawn to the possibilityof including, in this type of models, the risk accompanying the implementation of processes. Probability density functions that can beused to characterize random variables present in the model are presented. The possibility of making mistakes while creat-ing this typeof models was pointed out. Two selected examples of the use of stochastic simulation in the analysis of production processes on theexample of the mining process are presented.

scholarly journals Generalized Moment Generating Functions of Random Variables and Their Probability Density Functions

2017 ◽  
Vol 5 (2) ◽  
pp. 49-53 ◽  
Author(s):  
Matthew Chukwuma Michael ◽  
Oyeka Cyprain Anene ◽  
Ashinze Mpuruoma Akudo ◽  
Igabari John Nwabueze
Keyword(s):  
Probability Density ◽  
Generating Functions ◽  
Random Variables ◽  
Probability Density Functions ◽  
Density Functions ◽  
Moment Generating Functions ◽  
Generalized Moment

Classes of probability density functions having Laplace transforms with negative zeros and poles

1987 ◽  
Vol 19 (03) ◽  
pp. 632-651 ◽  
Author(s):  
Ushio Sumita ◽  
Yasushi Masuda
Keyword(s):  
Probability Density ◽  
Practical Importance ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Laplace Transforms ◽  
Probability Density Functions ◽  
Independent Random Variables ◽  
Density Functions ◽  
Exponential Random Variables ◽  
Zeros And Poles

We consider a class of functions on [0,∞), denoted by Ω, having Laplace transforms with only negative zeros and poles. Of special interest is the class Ω+ of probability density functions in Ω. Simple and useful conditions are given for necessity and sufficiency of f ∊ Ω to be in Ω+. The class Ω+ contains many classes of great importance such as mixtures of n independent exponential random variables (CMn), sums of n independent exponential random variables (PF∗ n ), sums of two independent random variables, one in CMr and the other in PF ∗ 1 (CMPFn with n = r + l) and sums of independent random variables in CMn (SCM). Characterization theorems for these classes are given in terms of zeros and poles of Laplace transforms. The prevalence of these classes in applied probability models of practical importance is demonstrated. In particular, sufficient conditions are given for complete monotonicity and unimodality of modified renewal densities.

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