The Poisson theorem for Bernoulli trials with a random probability of success and a discrete analog of the Weibull distribution

A problem related to the Bernoulli trials with a random probability of success is considered. First, as a result of the preliminary experiment, the value of the random variable $\pi\in (0,1)$ is determined that is taken as the probability of success in the Bernoulli trials. Then, the random variable $N$ is determined as the number of successes in $k\in\mathbb{N}$ Bernoulli trials with the so determined success probability $\pi$. The distribution of the random variable $N$ iscalled $\pi$-mixed binomial. Within the framework of these Bernoulli trials with the random probability of success, a “random” analog of the classical Poisson theorem is formulated for the $\pi$-mixed binomial distributions, in which the limit distribution turns out to be the mixed Poisson distribution. Special attention is paid to the case where mixing is performed with respect to the Weibull distribution. The corresponding mixed Poisson distribution called Poisson–Weibull law is proposed as a discrete analog of the Weibull distribution. Some properties of the Poisson–Weibull distribution are discussed. In particular, it is shown that this distribution can be represented as the mixed geometric distribution. A two-stage grid algorithm is proposed for estimation of parameters of mixed Poisson distributions and, in particular, of the Poisson–Weibull distribution. Statistical estimators for the upper bound of the grid are constructed. The examples of practical computations performed by the proposed algorithm are presented.