The **gamma-normal distribution** (also called the *Gaussian normal distribution*, *GN distribution* or *normal-gamma distribution*) is a bivariate compound probability distribution for the sum of a gamma random variable and a normal/Gaussian random variable. It is the conjugate prior (from Bayesian statistics) of a normal distribution with unknown mean and precision [1]. “Conjugate” here means that the surface is a gamma-normal both before and after historical results have been taken into account. As you collect more sample data, the parameters change and therefore the shape of the surface changes as well.

The distribution has four parameters: mu (μ), kappa (κ, alpha α, and beta β. The distribution can describe a multitude of surface shapes, depending on these four values (a.k.a. parameters).

If random variables X, Y, have a normal-gamma distribution, then the marginal distribution of Y is a gamma distribution with parameters α, β. The conditional distribution of X, given Y, is a normal distribution with expected value μ and logged standard deviation κ -log(Y)/2.

## Probability Density Function for the Gamma-Normal Distribution

If X ~ N(μ, σ^{2}) and σ ~ inverse gamma (α, β), then the PDF of the random variable X is given by [2]:

For -∞ x ∞.

## References

[1] Bernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory, Wiley. ISBN 0-471-49464-X

[2] Theorem. Retrieved December 7, 2021 from: http://www.math.wm.edu/~leemis/chart/UDR/PDFs/NormalGammanormal.pdf