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The gamma distribution is the conjugate prior for the precision of the normal distribution with known mean. 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 variance-gamma distribution, generalized Laplace distribution [2] or Bessel function distribution [2] is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the gamma distribution. The tails of the distribution decrease more slowly than the normal distribution. It is ...
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
A gamma distribution with shape parameter α = v/2 and rate parameter β = 1/2 is a chi-squared distribution with ν degrees of freedom. A chi-squared distribution with 2 degrees of freedom (k = 2) is an exponential distribution with a mean value of 2 (rate λ = 1/2 .)
meaning that the conditional distribution is a normal distribution with mean and precision — equivalently, with variance / (). Suppose also that the marginal distribution of T is given by T ∣ α , β ∼ Gamma ( α , β ) , {\displaystyle T\mid \alpha ,\beta \sim \operatorname {Gamma} (\alpha ,\beta ),}
The line is the trend in the mean. The plot demonstrates that the variance is not constant. The smoothed conditional variance against the smoothed conditional mean. The quadratic shape is indicative of the Gamma Distribution. The variance function of a Gamma is V() =
The marginal distribution of a gamma process at time is a gamma distribution with mean / and variance /. That is, the probability distribution f {\displaystyle f} of the random variable X t {\displaystyle X_{t}} is given by the density f ( x ; t , γ , λ ) = λ γ t Γ ( γ t ) x γ t − 1 e − λ x . {\displaystyle f(x;t,\gamma ,\lambda ...
The off-diagonal element is less familiar but can be identified as a normal variance-mean mixture where the mixing density is a χ 2 distribution. The corresponding marginal probability density for the off-diagonal element is therefore the variance-gamma distribution