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Concept
Gamma distribution
Summary
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use:
With a shape parameter and a scale parameter .
With a shape parameter and an inverse scale parameter , called a rate parameter.
In each of these forms, both parameters are positive real numbers.
The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and a base measure) for a random variable for which E[X] = kθ = α/β is fixed and greater than zero, and E[ln(X)] = ψ(k) + ln(θ) = ψ(α) − ln(β) is fixed (ψ is the digamma function).
The parameterization with k and θ appears to be more common in econometrics and other applied fields, where 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. See Hogg and Craig for an explicit motivation.
The parameterization with and is more common in Bayesian statistics, where the gamma distribution is used as a conjugate prior distribution for various types of inverse scale (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 a positive 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.
The gamma distribution can be parameterized in terms of a shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter.
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Related concepts (49)
Chi-squared distribution
In probability theory and statistics, the chi-squared distribution (also chi-square or -distribution) with degrees of freedom is the distribution of a sum of the squares of independent standard normal random variables. The chi-squared distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in construction of confidence intervals.
Erlang distribution
The Erlang distribution is a two-parameter family of continuous probability distributions with support . The two parameters are: a positive integer the "shape", and a positive real number the "rate". The "scale", the reciprocal of the rate, is sometimes used instead. The Erlang distribution is the distribution of a sum of independent exponential variables with mean each. Equivalently, it is the distribution of the time until the kth event of a Poisson process with a rate of .
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