In this paper, the two-observational  percentile, percentile and maximum likelihood Estimation of the Probability Density Function of  Inverse Weibull random variable are studied. Finally, these estimates are compared using simulation studies and a real data.

Membership Function based on Probability Density Functionچندین روش به منظور برآورد تابع چگالی احتمال وجود دارد. از سوی دیگر، در نظریه مجموعه هایفازی یکی از روش های ساختن تابع عضویت بر پایه ی مجموعه داده، روش مبتنی بر تابع چگالی احتمال است.با توجه به روش های متداول در برآورد تابع چگالی، این مسئله می تواند به محاسبه انواع تابع عضویت بر پایهیک مجموعه داده منجر شود. در این مقاله، برخی از این روش ها بیان و با مثال عددی تشریح می شوند..

Although knowing the time of the occurrence of the earthquakes is vital and helpful, unfortunately it is still unpredictable. By the way there is an urgent need to find a method to foresee this catastrophic event. There are a lot of methods for forecasting the time of earthquake occurrence. Another method for predicting that is to know Probability Density Function of time interval between earthquakes. In this paper a new Probability Density Function (PDF) for the time interval between earthquakes is found out. The parameters of the PDF will be estimated, and ultimately, the PDF will be tested by the earthquakes data about Iran.

Kernel method is one of the most common nonparametric Density Estimation and recently B-spline is used for Estimation of a Probability Density Function. These two methods in some how depend on selecting a smoothing parameter that has an important effect on precision of the estimators. In this paper, we consider kernel and B-spline methods of Density Estimation and smoothing parameter selection for these two methods. Then, the accuracy of the obtained estimators is compared by their mean square errors. Also, the effect of the number and dispersion of data on precision of estimators are studied. The results show that for a symmetric Probability Density, if the dispersion of data increases, the precision of both estimators decreases, while, for an asymmetric Probability Density Function, the precision of the estimators increases for dispersion data.

Probability Density Functions of the involved random variables are essential for the reliability-based design of offshore structures. The objective of present research was the derivation of Probability Density Function (PDF) for the local joint flexibility (LJF) factor, fLJF, in two-planar tubular DK-joints commonly found in jacket-type offshore structures. A total of 162 finite element (FE) analyses were carried out on 81 FE models of DK-joints subjected to two types of axial loading. Generated FE models were validated using available experimental data, FE results, and design formulas. Based on the results of parametric FE study, a sample database was prepared for the fLJF values and Density histograms were generated for respective samples based on the Freedman-Diaconis rule. Nine theoretical PDFs were fitted to the developed histograms and the maximum likelihood (ML) method was applied to evaluate the parameters of fitted PDFs. In each case, the Kolmogorov-Smirnov and chi-squared tests were used to evaluate the goodness of fit. Finally, the Inverse Gaussian model was proposed as the governing Probability distribution Function for the fLJF. After substituting the values of estimated parameters, two fully defined PDFs were presented for the fLJF in tubular DK-joints subjected to two types of axial loading.

Urban runoff has a large rate of various pollutants and destroys acceptor water resources and also damage environment. The management strategies for urban runoff pollutants controlling depend on identifying and analyzing these pollutants. In this study the procedure for fitting, selecting and appropriate Probability distribution Function (pdf) for a runoff quality constituent concentration in Isfahan was investigated. The best fitted probable distribution of pollutants concentration was determined and it was used for Estimation of concentrations with different return periods. The gtimated pollutant concentration with different return periods is using Selected pdf The results showed BOD5 and COD can be modeled by Gamma distribution, and TSS, No3, TP, Ph are statistically fitted with 2 lognormal distribution. Also, pollutant concentrations were increased with return period.

In this paper, the maximum liklihood Estimation, unbiased Estimations with minimum variance, percentile Estimation, best percentile Estimation single-observation Estimation and the best percentile Estimation two-observations in class which are based on order statistics are calculated in two sections for Probability Density and cumulative distribution Functions of the beta Weibull geometric distribution, specially with bathtub-shaped and unimodal failure rate which are useful for modeling of data related to reliability and lifetime. Furthermore, through the simulation method of Monte Carlo and calculation of average square of errors of estimators, they are subjected to comparisons ultimately, the desirable estimator in each section is determined.