A new multivariate model involving geometric sums and maxima of exponentials

We study the joint distribution of (X, Y, N), where N has a geometric distribution while X and Y are, respectively, the sum and the maximum of N i.i.d. exponential random variables. We present fundamental properties of this class of mixed trivariate distributions, and discuss their applications. Our results include explicit formulas for the marginal and conditional distributions, joint integral transforms, moments and related parameters, stability properties, and stochastic representations. We also derive maximum likelihood estimators for the parameters of this distribution, along with their asymptotic properties, and briefly discuss certain generalizations of this model. An example from finance, where N represents the duration of the growth period of the daily log-returns of currency exchange rates, illustrates the modeling potential of this model.