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The gamma distribution is a two-parameter exponential family with natural parameters α − 1 and −1/θ (equivalently, α − 1 and −λ), and natural statistics X and ln X. If the shape parameter α is held fixed, the resulting one-parameter family of distributions is a natural exponential family.
In probability theory and statistics, a shape parameter (also known as form parameter) [1] is a kind of numerical parameter of a parametric family of probability distributions [2] that is neither a location parameter nor a scale parameter (nor a function of these, such as a rate parameter).
It is a generalization of the gamma distribution which has one shape parameter (and a scale parameter). Since many distributions commonly used for parametric models in survival analysis (such as the exponential distribution , the Weibull distribution and the gamma distribution ) are special cases of the generalized gamma, it is sometimes used ...
The Gamma distribution, which describes the time until n consecutive rare random events occur in a process with no memory. The Erlang distribution, which is a special case of the gamma distribution with integral shape parameter, developed to predict waiting times in queuing systems; The inverse-gamma distribution; The generalized gamma distribution
The inverse gamma distribution's probability density function is defined over the support > (;,) = (/) + (/)with shape parameter and scale parameter. [2] Here () denotes the gamma function.
A Weibull distribution is a generalized gamma distribution with both shape parameters equal to k. The translated Weibull distribution (or 3-parameter Weibull) contains an additional parameter. [ 12 ] It has the probability density function
A beta distribution with shape parameters α = β = 1 is a continuous uniform distribution over the real numbers 0 to 1. A beta-binomial distribution with parameter n and shape parameters α = β = 1 is a discrete uniform distribution over the integers 0 to n. A Student's t-distribution with one degree of freedom (v = 1) is a Cauchy ...
The Nakagami distribution or the Nakagami-m distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter / and a scale parameter >. It is used to model physical phenomena such as those found in medical ultrasound imaging, communications engineering ...