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The terms "distribution" and "family" are often used loosely: Specifically, an exponential family is a set of distributions, where the specific distribution varies with the parameter; [a] however, a parametric family of distributions is often referred to as "a distribution" (like "the normal distribution", meaning "the family of normal distributions"), and the set of all exponential families ...
Download as PDF; Printable version; ... Pages in category "Exponential family distributions" ... Kaniadakis exponential distribution;
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
For example, the Bernoulli distribution is a binomial distribution with n = 1 trial, the exponential distribution is a gamma distribution with shape parameter α = 1 (or k = 1 ), and the geometric distribution is a special case of the negative binomial distribution. Some exponential family distributions are not NEF.
Since the functions are the sufficient statistics of the Dirichlet distribution, the exponential family differential identities can be used to get an analytic expression for the expectation of (see equation (2.62) in [12]) and its associated covariance matrix:
The Dagum distribution; The exponential distribution, which describes the time between consecutive rare random events in a process with no memory. The exponential-logarithmic distribution; The F-distribution, which is the distribution of the ratio of two (normalized) chi-squared-distributed random variables, used in the analysis of variance.
The Bernoulli distributions for form an exponential family. The maximum likelihood estimator of p {\displaystyle p} based on a random sample is the sample mean . The probability mass distribution function of a Bernoulli experiment along with its corresponding cumulative distribution function.
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).