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The form of the density function of the Weibull distribution changes drastically with the value of k. For 0 < k < 1, ... where Γ is the gamma function.
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 ).
This arises because the ordinary Weibull distribution is used for cases that deal with data minima rather than data maxima. The distribution here has an addition parameter compared to the usual form of the Weibull distribution and, in addition, is reversed so that the distribution has an upper bound rather than a lower bound.
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 to determine which parametric model is appropriate for a given set of data. [1]
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 matrix gamma distribution and the Wishart distribution are multivariate generalizations of the gamma distribution (samples are positive-definite matrices rather than positive real numbers). The gamma distribution is a special case of the generalized gamma distribution , the generalized integer gamma distribution , and the generalized ...
The Fréchet distribution, also known as inverse Weibull distribution, [2] [3] is a special case of the generalized extreme value distribution. It has the cumulative distribution function ( ) = > . where α > 0 is a shape parameter.
They showed that the exponentiated Weibull distribution has increasing, decreasing, bathtub, and unimodal hazard rates. The exponentiated exponential distribution proposed by Gupta and Kundu (1999, 2001) is a special case of the exponentiated Weibull family. Later, the moments of the EW distribution were derived by Choudhury (2005).