This paper presents likelihood–based inference methods for the family of univariate gamma–normal distributions \(\textrm{GN}(\alpha , r, \mu , \sigma ^2)\) that result from summing independent \(\gamma (\alpha , r)\) and \(N(\mu ,\sigma ^2)\) random variables. First, the probability density function of a gamma–normal variable is provided in compact form with the use of parabolic cylinder functions, along with key properties. We then provide analytic expressions for the maximum–likelihood score equations and the Fisher information matrix, and discuss inferential methods for the gamma–normal distribution. Given the widespread use of the two constituting distributions, the gamma–normal distribution is a general purpose tool for a variety of applications. In particular, we discuss two distributions that are obtained as special cases and that are featured in a variety of statistical applications: the exponential–normal distribution and the chi–squared–normal (or overdispersed chi–squared) distribution.
