On the distribution of the product of two continuous random variables with an application to electricity market transactions. Finite and infinite-variance case

Abstract

In this paper we study the distribution of a product of two continuous random variables. We derive formulas for the probability density functions and moments of the products of the Gaussian, log-normal, Student's t and Pareto random variables. In all cases we analyze separately independent as well as correlated random variables. Based on the theoretical results we use the general maximum likelihood approach for the estimation of the parameters of the product random variables and apply the methodology for a real data case study. We analyze a distribution of the transaction values, being a product of prices and volumes, from a continuous trade on the German intraday electricity market.

Dependence structure for the product of bi-dimensional finite-variance VAR(1) model components. An application to the cost of electricity load prediction errors

“…It is worth mentioning that in the case when the residual series has bivariate Gaussian distribution, then X 1 (t) and X 2 (t) for each t ∈ Z have one-dimensional Gaussian distributions and Y (t) has a variancegamma distribution with appropriate parameters (see Aroian et al, 1978). A detailed analysis related to the distribution of a product Gaussian random variables was presented, for instance, by Adamska et al (2021). In case when (Z 1 , Z 2 ) have the bivariate Student's t distribution, the components of the VAR(1) model X 1 (t) and X 2 (t) are not Student's t distributed.…”

“…The distribution of the product time series is not deeply analyzed in this paper, however it is worth mentioning, that it can be obtained by using the results for the product random variables, see e.g. Adamska et al (2021).…”