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The term “inverse normal distribution” is informal and doesn’t refer to a specific probability distribution [1]. However, there is an* inverse normal density function *in statistics that reverses the procedure for finding z-values (see “Inverse Normal Distribution” below). There is also a ** generalized inverse normal distribution** family, proposed by Robert in 1991 [2].

## Robert’s generalized inverse normal distribution

Robert’s generalized inverse normal distribution has a probability density function (PDF) of

where *K(a*, ξ, σ) is the normalizing constant, which can be expressed explicitly in terms of confluent hypergeometric functions [3].

This is a bimodal distribution with two modes at

Robert described a second generalized inverse normal distribution as a generalization of the distribution of 1/*X*, when X is distributed as a normally distributed random variable with mean 0 and variance σ^{2}. The PDF of *Z* = 1/*X* is

The generalized inverse distribution has been used by subsequent authors including Druilhet and Pommeret in their paper on Bayesian analysis [4].

## Inverse Normal Distribution

The “inverse normal distribution” refers to the use of an inverse normal density function in statistics to **reverse the procedure for finding z-values**. Instead of taking a z-score and looking up the percentage area in a table, you take the percentage and look up the z-score. In this sense, the inverse normal distribution is *not *equal to the Inverse Gaussian distribution, which is a two-parameter family of continuous probability distributions. Perhaps confusingly, the “inverse” in “inverse Gaussian” is misleading because the distribution *isn’t actually an inverse*. At large values of its shape parameter, **the inverse Gaussian approximates the normal distribution.**

## References

[1] Stephanie Glen. “Inverse Normal Distribution” From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/inverse-normal-distribution/

[2] Robert, C. (1991). Generalized inverse normal distributions, Statistics & Probability Letters, 11, 37-41.

[3] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

[4] Druilet & Pommeret. (2012). Invariant Conjugate Analysis for Exponential Families. Bayesian Analysis 7, Number 4, pp. 903–916