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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 > fixed, it is also a single-parameter natural exponential family distribution [3] where the base distribution has density
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 .
The cumulative distribution function (shown as F(x)) gives the p values as a function of the q values. The quantile function does the opposite: it gives the q values as a function of the p values. Note that the portion of F(x) in red is a horizontal line segment.
The inverse cumulative distribution function (quantile function) of the logistic distribution is a generalization of the logit function. Its derivative is called the quantile density function. They are defined as follows: (;,) = + ().
The Q-function can be generalized to higher dimensions: [14] = (),where (,) follows the multivariate normal distribution with covariance and the threshold is of the form = for some positive vector > and positive constant >.
The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing [41] (Matlab code). In the following sections we look at some special cases.
The cumulative distribution function ... The quantile function (or inverse CDF) is written: ... Rectified Gaussian distribution; References
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 ...