In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success or failure. A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events.
In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of successes occurs. For example, we can define rolling a 6 on some dice as a success, and rolling any other number as a failure, and ask how many failure rolls will occur before we see the third success. In such a case, the probability distribution of the number of failures that appear will be a negative binomial distribution.
A hidden Markov model (HMM) is a Markov model in which the observations are dependent on a latent Markov process. An HMM requires that there be an observable process whose outcomes depend on the outcomes of in a known way. Since cannot be observed directly, the goal is to learn about state of by observing By definition of being a Markov model, an HMM has an additional requirement that the outcome of at time must be "influenced" exclusively by the outcome of at and that the outcomes of and at must be conditionally independent of at given at time Estimation of the parameters in an HMM can be performed using maximum likelihood. For linear chain HMMs, the Baum–Welch algorithm can be used to estimate the parameters.
In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of successes 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 statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value.
In probability and statistics, the Dirichlet distribution, often denoted , is a family of continuous multivariate probability distributions parameterized by a vector of positive reals. It is a multivariate generalization of the beta distribution, hence its alternative name of multivariate beta distribution (MBD). Dirichlet distributions are commonly used as prior distributions in Bayesian statistics, and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution.
In the field of information retrieval, divergence from randomness, one of the first models, is one type of probabilistic model. It is basically used to test the amount of information carried in the documents. It is based on Harter's 2-Poisson indexing-model. The 2-Poisson model has a hypothesis that the level of the documents is related to a set of documents which contains words occur relatively greater than the rest of the documents. It is not a 'model', but a framework for weighting terms using probabilistic methods, and it has a special relationship for term weighting based on notion of eliteness.
In decision theory, the Ellsberg paradox is a paradox in which people's decisions are inconsistent with subjective expected utility theory. John Maynard Keynes published a version of the paradox in 1921. Daniel Ellsberg popularized the paradox in his 1961 paper, "Risk, Ambiguity, and the Savage Axioms". It is generally taken to be evidence of ambiguity aversion, in which a person tends to prefer choices with quantifiable risks over those with unknown, incalculable risks.
Lottery mathematics is used to calculate probabilities of winning or losing a lottery game. It is based primarily on combinatorics, particularly the twelvefold way and combinations without replacement.
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.
In probability theory, Dirichlet processes are a family of stochastic processes whose realizations are probability distributions. In other words, a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. It is often used in Bayesian inference to describe the prior knowledge about the distribution of random variables—how likely it is that the random variables are distributed according to one or another particular distribution.
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. 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. The Dirichlet parameter vector captures the prior belief about the situation and can be seen as a pseudocount: observations of each outcome that occur before the actual data is collected. The compounding corresponds to a Pólya urn scheme. It is frequently encountered in Bayesian statistics, machine learning, empirical Bayes methods and classical statistics as an overdispersed multinomial distribution.
In statistics, the hypergeometric distribution is the discrete probability distribution generated by picking colored balls at random from an urn without replacement.
In probability theory and statistics, Wallenius' noncentral hypergeometric distribution is a generalization of the hypergeometric distribution where items are sampled with bias.
In probability theory and statistics, Fisher's noncentral hypergeometric distribution is a generalization of the hypergeometric distribution where sampling probabilities are modified by weight factors. It can also be defined as the conditional distribution of two or more binomially distributed variables dependent upon their fixed sum.
In statistics, a Pólya urn model, named after George Pólya, is a family of urn models that can be used to interpret many commonly used statistical models.
In probability theory, a beta negative binomial distribution is the probability distribution of a discrete random variable equal to the number of failures needed to get successes in a sequence of independent Bernoulli trials. The probability of success on each trial stays constant within any given experiment but varies across different experiments following a beta distribution. Thus the distribution is a compound probability distribution.
In probability theory and statistics, the negative hypergeometric distribution describes probabilities for when sampling from a finite population without replacement in which each sample can be classified into two mutually exclusive categories like Pass/Fail or Employed/Unemployed. As random selections are made from the population, each subsequent draw decreases the population causing the probability of success to change with each draw. Unlike the standard hypergeometric distribution, which describes the number of successes in a fixed sample size, in the negative hypergeometric distribution, samples are drawn until failures have been found, and the distribution describes the probability of finding successes in such a sample. In other words, the negative hypergeometric distribution describes the likelihood of successes in a sample with exactly failures.
