In probability theory and statistics, the beta-binomial distribution is a family of discrete probability distributions on a finite support of non-negative integers arising when the probability of success in each of a fixed or known number of Bernoulli trials is either unknown or random. The beta-binomial distribution is the binomial distribution in which the probability of success at each of n trials is not fixed but randomly drawn from a beta distribution. It is frequently used in Bayesian statistics, empirical Bayes methods and classical statistics to capture overdispersion in binomial type distributed data.
The beta-binomial is a one-dimensional version of the Dirichlet-multinomial distribution as the binomial and beta distributions are univariate versions of the multinomial and Dirichlet distributions respectively. The special case where α and β are integers is also known as the negative hypergeometric distribution.
The Beta distribution is a conjugate distribution of the binomial distribution. This fact leads to an analytically tractable compound distribution where one can think of the parameter in the binomial distribution as being randomly drawn from a beta distribution.
Suppose we were interested in predicting the number of heads, in future trials. This is given by
Using the properties of the beta function, this can alternatively be written
The beta-binomial distribution can also be motivated via an urn model for positive integer values of α and β, known as the Pólya urn model. Specifically, imagine an urn containing α red balls and β black balls, where random draws are made. If a red ball is observed, then two red balls are returned to the urn. Likewise, if a black ball is drawn, then two black balls are returned to the urn. If this is repeated n times, then the probability of observing x red balls follows a beta-binomial distribution with parameters n, α and β.
If the random draws are with simple replacement (no balls over and above the observed ball are added to the urn), then the distribution follows a binomial distribution and if the random draws are made without replacement, the distribution follows a hypergeometric distribution.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
The objective of this course is to provide a detailed coverage of the standard models for the valuation and hedging of derivatives products such as European options, American options, forward contract
This course presents an introduction to statistical mechanics geared towards materials scientists. The concepts of macroscopic thermodynamics will be related to a microscopic picture and a statistical
In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of successes (random draws for which the object drawn has a specified feature) in draws, without replacement, from a finite population of size that contains exactly objects with that feature, wherein each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of successes in draws with replacement.
In probability and statistics, an urn problem is an idealized mental exercise in which some objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. One pretends to remove one or more balls from the urn; the goal is to determine the probability of drawing one color or another, or some other properties. A number of important variations are described below.
In probability theory and statistics, the Dirichlet-multinomial distribution is a family of discrete multivariate probability distributions on a finite support of non-negative integers. It is also called the Dirichlet compound multinomial distribution (DCM) or multivariate Pólya distribution (after George Pólya). It is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector , and an observation drawn from a multinomial distribution with probability vector p and number of trials n.
Covers the binomial model for asset pricing, including options pricing and convergence to the Black-Scholes model.
Explores counting bit strings, committees, balls distribution, poker hands, and coefficients, with examples of the Pigeonhole Principle and card selection.
Demonstrates dry grinding using a rotary ball mill and emphasizes safety measures.
The effects of operational and geometrical uncertainties on Tip Leakage Vortex (TLV) characteristics are investigated in the current research. Geometrical uncertainties are comprised of manufacturing tolerances or gradual geometry degradation over the time ...
Spatial count data models are used to explain and predict the frequency of phenomena such as traffic accidents in geographically distinct entities such as census tracts or road segments. These models are typically estimated using Bayesian Markov chain Mont ...
In this thesis, we investigate the inverse problem of trees and barcodes from a combinatorial, geometric, probabilistic and statistical point of view.Computing the persistent homology of a merge tree yields a barcode B. Reconstructing a tree from B involve ...