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Probability Distributions01:32

Probability Distributions

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 The probability of a random variable x  is the likelihood of its occurrence. A probability distribution represents the probabilities of a random variable using a formula, graph, or table. There are two types of probability distribution– discrete probability distribution and continuous probability distribution.
A discrete probability distribution is a probability distribution of discrete random variables. It can be categorized into binomial probability distribution and Poisson...
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Applications of Normal Distribution01:22

Applications of Normal Distribution

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The normal distribution is a useful statistical tool. One of its practical applications is determining the door height after considering the normal distribution of heights of persons, such that many can pass through it easily without striking their heads. The normal distribution can also determine the probability of a person having a height less than a specific height.
The heights of 15 to 18-year-old males from Chile from 1984 to 1985 followed a normal distribution. The mean height is 172.36...
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Parametric Survival Analysis: Weibull and Exponential Methods01:14

Parametric Survival Analysis: Weibull and Exponential Methods

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Parametric survival analysis models survival data by assuming a specific probability distribution for the time until an event occurs. The Weibull and exponential distributions are two of the most commonly used methods in this context, due to their versatility and relatively straightforward application.
Weibull Distribution
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Distributions to Estimate Population Parameter01:26

Distributions to Estimate Population Parameter

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The accurate values of population parameters such as population proportion, population mean, and population standard deviation (or variance) are usually unknown. These are fixed values that can only be estimated from the data collected from the samples. The estimates of each of these parameters are sample proportion, the sample mean, and sample standard deviation (or variance). To obtain the values of these sample statistics, data are required that have particular distribution and central...
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Binomial Probability Distribution01:15

Binomial Probability Distribution

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A binomial distribution is a probability distribution for a procedure with a fixed number of trials, where each trial can have only two outcomes.
The outcomes of a binomial experiment fit a binomial probability distribution. A statistical experiment can be classified as a binomial experiment if the following conditions are met:
There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials.
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Normal Distribution01:11

Normal Distribution

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The normal, a continuous distribution, is the most important of all the distributions. Its graph is a bell-shaped symmetrical curve, which is observed in almost all disciplines. Some of these include psychology, business, economics, the sciences, nursing, and, of course, mathematics. Some instructors may use the normal distribution to help determine students’ grades. Most IQ scores are normally distributed. Often real-estate prices fit a normal distribution. The normal distribution is...
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Related Experiment Video

Updated: Sep 21, 2025

Controlled Synthesis and Fluorescence Tracking of Highly Uniform PolyN-isopropylacrylamide Microgels
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A new extended gumbel distribution: Properties and application.

Aisha Fayomi1, Sadaf Khan2, Muhammad Hussain Tahir2

  • 1Faculty of Science, Department of Statistics, King Abdulaziz University, Jeddah, Saudi Arabia.

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|May 27, 2022
PubMed
Summary

A new generalized Gumbel distribution family, based on the T-X paradigm, is introduced. This robust statistical model offers enhanced properties for risk assessment and data analysis.

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Area of Science:

  • Statistics
  • Probability Theory
  • Mathematical Modeling

Background:

  • The Gumbel distribution is a fundamental tool in extreme value theory.
  • Generalizations of existing distributions are crucial for expanding their applicability.
  • The T-X paradigm offers a flexible framework for creating new probability distributions.

Purpose of the Study:

  • To propose a robust generalization of the Gumbel distribution using the T-X paradigm.
  • To derive and analyze the statistical properties of the new distribution family.
  • To demonstrate the model's utility through simulation and real-world data applications.

Main Methods:

  • Development of a new distribution family based on a linear combination of generalized exponential distributions.
  • Derivation of exact formulations for key statistical features: quantile function, moments, generating function, and order statistics.
  • Parameter estimation using the maximum likelihood method and skewness analysis including MacGillivray skewness.

Main Results:

  • A novel generalized Gumbel distribution family and three specific models were successfully formulated.
  • Comprehensive statistical properties were derived, providing a thorough understanding of the distributions.
  • The proposed model demonstrated superiority and authenticity in real-world data modeling and simulation assessments.

Conclusions:

  • The proposed generalized Gumbel distribution family offers a valuable and flexible extension to existing models.
  • The derived statistical properties and estimation methods facilitate practical applications.
  • The model's performance in risk indicator determination and data fitting highlights its potential in various fields.