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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 geometric distribution is the discrete probability distribution that describes when the first success in an infinite sequence of independent and identically distributed Bernoulli trials occurs. Its probability mass function depends on its parameterization and support. When supported on , the probability mass function is where is the number ...
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral. for complex number inputs such that . The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its ...
The reciprocal 1/ X of a random variable X, is a member of the same family of distribution as X, in the following cases: Cauchy distribution, F distribution, log logistic distribution. Examples: If X is a Cauchy (μ, σ) random variable, then 1/ X is a Cauchy (μ / C, σ / C) random variable where C = μ2 + σ2. If X is an F (ν1, ν2) random ...
t. e. Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. In these contexts, the capital letters and the small letters represent distinct and unrelated entities.
Polar coordinates at the crack tip. In fracture mechanics, the stress intensity factor (K) is used to predict the stress state ("stress intensity") near the tip of a crack or notch caused by a remote load or residual stresses. [1] It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing ...
where is the gamma function and is the hypergeometric function. 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 ...
v. t. e. In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, [1] is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability . Less formally, it can be thought of as a model for the set of possible outcomes ...