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The inverse Gaussian distribution has several properties analogous to a Gaussian distribution. The name can be misleading: it is an "inverse" only in that, while the Gaussian describes a Brownian motion's level at a fixed time, the inverse Gaussian describes the distribution of the time a Brownian motion with positive drift takes to reach a ...
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
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 ...
Cumulative distribution function for the exponential distribution Cumulative distribution function for the normal distribution. In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable, or just distribution function of , evaluated at , is the probability that will take a value less than or equal to .
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In probability theory and statistics, an inverse distribution is the distribution of the reciprocal of a random variable. Inverse distributions arise in particular in the Bayesian context of prior distributions and posterior distributions for scale parameters.
The probability density, cumulative distribution, and inverse cumulative distribution functions of a generalized chi-squared variable do not have simple closed-form expressions. But there exist several methods to compute them numerically: Ruben's method, [ 7 ] Imhof's method, [ 8 ] IFFT method, [ 6 ] ray method, [ 6 ] and ellipse approximation.
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 +.