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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).
A binomial test is a statistical hypothesis test used to determine whether the proportion of successes in a sample differs from an expected proportion in a binomial distribution. It is useful for situations when there are two possible outcomes (e.g., success/failure, yes/no, heads/tails), i.e., where repeated experiments produce binary data.
This can now be considered a binomial distribution with = trial, so a binary regression is a special case of a binomial regression. If these data are grouped (by adding counts), they are no longer binary data, but are count data for each group, and can still be modeled by a binomial regression; the individual binary outcomes are then referred ...
The definition of the Clopper–Pearson interval can also be modified to obtain exact confidence intervals for different distributions. For instance, it can also be applied to the case where the samples are drawn without replacement from a population of a known size, instead of repeated draws of a binomial distribution.
The concept of data type is similar to the concept of level of measurement, but more specific. For example, count data requires a different distribution (e.g. a Poisson distribution or binomial distribution) than non-negative real-valued data require, but both fall under the same level of measurement (a ratio scale).
Binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of independent occurrences; Negative binomial distribution, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs
binary data Data that can take only two values, usually represented by the binary digits 0 and 1. binomial distribution bivariate analysis A type of quantitative statistical analysis in which exactly two variables are analyzed, for the purpose of determining the empirical relationship between them. Contrast multivariate analysis. blocking
When k = 2, the multinomial distribution is the binomial distribution. Categorical distribution, the distribution of each trial; for k = 2, this is the Bernoulli distribution. The Dirichlet distribution is the conjugate prior of the multinomial in Bayesian statistics. Dirichlet-multinomial distribution. Beta-binomial distribution.