The Pareto-Gamma Mixture

In this paper we will review some established properties and derive some new properties of a Pareto distribution with fixed scale whose unknown shape parameter is Gamma distributed. Namely:

• that Gamma is a conjugate prior to the Pareto distribution.
• the formula for the posterior parameters of the Gamma given observed data.
• a closed form for the CDF of the Pareto-Gamma mixture.
• that the mean and all higher moments of the distribution are infinite.
• a formula for the moments of the limited expected value of a random variable following this distribution.
• the intractability of the closed form.