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The inverse Gaussian distribution is a two-parameter exponential family with natural parameters −λ/(2μ 2) and −λ/2, and natural statistics X and 1/X. For λ > 0 {\displaystyle \lambda >0} fixed, it is also a single-parameter natural exponential family distribution [ 2 ] where the base distribution has density
The quantile function, Q, of a probability distribution is the inverse of its cumulative distribution function F. The derivative of the quantile function, namely the quantile density function, is yet another way of prescribing a probability distribution. It is the reciprocal of the pdf composed with the quantile function.
The cumulative distribution function (cdf) of the standard distribution is a scaled and shifted version of the Gudermannian function, = + () = ().where "arctan" is the inverse (circular) tangent function.
The class of normal-inverse Gaussian distributions is closed under convolution in the following sense: [9] if and are independent random variables that are NIG-distributed with the same values of the parameters and , but possibly different values of the location and scale parameters, , and ,, respectively, then + is NIG-distributed with parameters ,, + and +.
However, a Lévy process that is generalised hyperbolic at one point in time might fail to be generalized hyperbolic at another point in time. In fact, the generalized Laplace distributions and the normal inverse Gaussian distributions are the only subclasses of the generalized hyperbolic distributions that are closed under convolution. [4]
If Y has a half-normal distribution, then (Y/σ) 2 has a chi square distribution with 1 degree of freedom, i.e. Y/σ has a chi distribution with 1 degree of freedom. The half-normal distribution is a special case of the generalized gamma distribution with d = 1, p = 2, a = 2 σ {\displaystyle {\sqrt {2}}\sigma } .
[1] [2] In other words, () is the probability that a normal (Gaussian) random variable will obtain a value larger than standard deviations. Equivalently, Q ( x ) {\displaystyle Q(x)} is the probability that a standard normal random variable takes a value larger than x {\displaystyle x} .
The inverse Gaussian and gamma distributions are special cases of the generalized inverse Gaussian distribution for p = −1/2 and b = 0, respectively. [7] Specifically, an inverse Gaussian distribution of the form