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In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
The Beta distribution on [0,1], a family of two-parameter distributions with one mode, of which the uniform distribution is a special case, and which is useful in estimating success probabilities. The four-parameter Beta distribution, a straight-forward generalization of the Beta distribution to arbitrary bounded intervals [,].
Beta prime. In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind [1]) is an absolutely continuous probability distribution. If has a beta distribution, then the odds has a beta prime distribution.
Some distributions have been specially named as compounds: beta-binomial distribution, Beta negative binomial distribution, gamma-normal distribution. Examples: If X is a Binomial(n,p) random variable, and parameter p is a random variable with beta(α, β) distribution, then X is distributed as a Beta-Binomial(α,β,n).
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 ...
Definition. A generalized beta random variable, Y, is defined by the following probability density function: and zero otherwise. Here the parameters satisfy , and , , and positive. The function B ( p,q) is the beta function. The parameter is the scale parameter and can thus be set to without loss of generality, but it is usually made explicit ...
t. e. In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. [1] [2] It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events ( subsets of the sample space).
Diagram of the Pearson system, showing distributions of types I, III, VI, V, and IV in terms of β 1 (squared skewness) and β 2 (traditional kurtosis) The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of ...