Gamma Distribution

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. There are three different parameterizations in common use:

  1. With a shape parameter k and a scale parameter θ.
  2. With a shape parameter α = k and an inverse scale parameter β = 1⁄θ, called a rate parameter.
  3. With a mean parameter and the shape parameter k.

The parameterization with k and θ appears to be more common in econometrics and certain other applied fields, where e.g. the gamma distribution is frequently used to model waiting times. For instance, in life testing, the waiting time until death is a random variable that is frequently modeled with a gamma distribution.

The parameterization with α and β is more common in Bayesian statistics, where the gamma distribution is used as a conjugate prior distribution for various types of inverse scale (aka rate) parameters, such as the λ of an exponential distribution or a Poisson distribution — or for that matter, the β of the gamma distribution itself. (The closely related inverse gamma distribution is used as a conjugate prior for scale parameters, such as the variance of a normal distribution.)

If k is an integer, then the distribution represents an Erlang distribution; i.e., the sum of k independent exponentially distributed random variables, each of which has a mean of θ (which is equivalent to a rate parameter of 1/θ).

The gamma distribution is the maximum entropy probability distribution for a random variable X for which is fixed and greater than zero, and is fixed ( is the digamma function).

Read more about Gamma Distribution:  Characterization Using Shape k and Scale θ, Characterization Using Shape α and Rate β, Generating Gamma-distributed Random Variables, Applications

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... The Beta prime distribution The Birnbaum–Saunders distribution The chi distribution The noncentral chi distribution The chi-squared distribution ... It is a special case of the Gamma distribution, and it is used in goodness-of-fit tests in statistics ... The inverse-chi-squared distribution The noncentral chi-squared distribution The Scaled-inverse-chi-squared distribution The Dagum distribution The exponential ...
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... This section requires expansion The gamma distribution has been used to model the size of insurance claims and rainfalls ... accumulated in a reservoir are modelled by a gamma process ... The gamma distribution is also used to model errors in multi-level Poisson regression models, because the combination of the Poisson distribution and a gamma ...
Normal-gamma Distribution
... In probability theory and statistics, the normal-gamma distribution (or Gaussian-gamma distribution) is a bivariate four-parameter family of continuous ... It is the conjugate prior of a normal distribution with unknown mean and precision ...
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... The five most important univariate cases are normal distribution with known variance Poisson distribution gamma distribution with known shape parameter α (or k depending on notation set used) binomial ... Distributions such as the exponential, chi-squared, Rayleigh, Weibull, Bernoulli, and geometric distributions are special cases of the above five distributions ... Many common distributions are either NEF or can be related to the NEF ...
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... The normal distribution with fixed variance is NEF-QVF because the variance is constant ... The Poisson distribution is NEF-QVF because all Poisson distributions have variance equal to the mean, so variance is a linear function of the mean ... The Gamma distribution is NEF-QVF because the mean of the Gamma distribution is and the variance of the Gamma distribution is, so the variance is a quadratic function ...

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