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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 ...
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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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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
The distribution is a special case of the folded normal distribution with μ = 0.; It also coincides with a zero-mean normal distribution truncated from below at zero (see truncated normal distribution)
The cumulative distribution function of the reciprocal, within the same range, is G ( y ) = b − y − 1 b − a . {\displaystyle G(y)={\frac {b-y^{-1}}{b-a}}.} For example, if X is uniformly distributed on the interval (0,1), then Y = 1 / X has density g ( y ) = y − 2 {\displaystyle g(y)=y^{-2}} and cumulative distribution function G ( y ...
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.