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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 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 Dirichlet distribution, a generalization of the beta distribution. The Ewens's sampling formula is a probability distribution on the set of all partitions of an integer n, arising in population genetics. The Balding–Nichols model; The multinomial distribution, a generalization of the binomial distribution.
If n and m are large compared to N, and p = m/N is not close to 0 or 1, then X approximately has a Binomial(n, p) distribution. X is a beta-binomial random variable with parameters (n, α, β). Let p = α/(α + β) and suppose α + β is large, then X approximately has a binomial(n, p) distribution.
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 (with probability p) or failure (with probability q = 1 − p).
Beta distribution, for a single probability (real number between 0 and 1); conjugate to the Bernoulli distribution and binomial distribution; Gamma distribution, for a non-negative scaling parameter; conjugate to the rate parameter of a Poisson distribution or exponential distribution, the precision (inverse variance) of a normal distribution, etc.
Compounding a binomial distribution with probability of success distributed according to a beta distribution yields a beta-binomial distribution. It possesses three parameters, a parameter n {\displaystyle n} (number of samples) from the binomial distribution and shape parameters α {\displaystyle \alpha } and β {\displaystyle \beta } from the ...
The binomial distribution generalizes this to the number of heads from performing n independent flips (Bernoulli trials) of the same coin. The multinomial distribution models the outcome of n experiments, where the outcome of each trial has a categorical distribution, such as rolling a k-sided die n times. Let k be a fixed finite number.