Joint Distribution of Minimum of N Iid Exponential Random Variables and Poisson Marginal

We introduced a random vector (X,N), where N has Poisson distribution and X are minimum of N independent and identically distributed exponential random variables. We present fundamental properties of this vector such as PDF, CDF and stochastic representations. Our results include explicit formulas for marginal and conditional distributions, moments and moments generating functions. We also derive moments estimators and maximum likelihood estimators of the parameter of this distribution.