Abstract
In this study, we present a univariate probability distribution through application of the three Sub and Super Exponential heavier-longer and lighter-shorter tails fitting. This univariate family includes the Lognormal, Gamma and Weibull distribution, the adequacy of the distribution tails is obtained by adequate Fitting Tests and descriptive Criterion. It emphasizes on tail values and is independent of the number of intervals. In this regards the time series analysis for the last three centuries of the logarithm population data sets over to Karachi region (from1729 to1946 and from 1951 to 2018) is used, which contains irregular and regular length and peaks, That peaks /tails fitting is attained by methods for validation and normality tests and defined by stochastic depiction. In other hand, Weibull and Lognormal distribution tails are found as heavier distribution by two validation tests (Maximum Likelihood Estimation and probability of correct selection), In the final section, the univariate probability distributions are used to Monte Carlo simulation for generating the actual population data, it indicates that the heavy-tailed Lognormal and Weibull distributions are also fitted contract than the more commonly seen lighter tailed Gamma distribution. So, the Monte Carlo Simulation performs the appropriate Lognormal and Weibull distributions for irregular and regular data and generate data values (298 and 69) from duration of 1729 to 2020 and 1951 to 2020.
 Copyright(c) The Author
Highlights
The Relationship of Univariate Probability Distributions is formed as a mathematical function, which provides the occurrence of probabilities in different outcomes in any random phenomenon
An AD test is better to the KS test, it stretches more perception evidence linked to the distribution tails, the forte of this test is not depend on the quantity of intervals, By using the selective univariate distributions and its tails analysis, more appropriate distributions are computed to comparing the adequate fitting test, less value of AD test suggest to suitability of tail fitting, as well two types of sub and super tail fitting are more applicable in our data intervals
We have investigated to concert of these three tails that analogous to the Gamma, Lognormal and Weibull distributions tail fitting, considering as subsuper exponential empirical tails, their appropriateness is based on validation methods (PDF, Probability of Correct Selection (PCS) and Maximum Likelihood Estimation (MLE))
Summary
The Relationship of Univariate Probability Distributions is formed as a mathematical function, which provides the occurrence of probabilities in different outcomes in any random phenomenon. The set of real numbers of sample space is known as univariate probability distribution [18, 27]. A univariate probability distribution is used to disperse a probability to several outcomes of a random phenomenon. When the set of completely possible outcomes to any random phenomenon is countable, the probability distribution can be defined in a Probability Mass Function The univariate probability distribution of the mix two cases is called as Mixed DiscreteContinuous distribution. Weibull (αα, ββ) MMββ, SS, VV ηη : Integer Parameters P: condition 0< p
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
