# Transmuted distributions and extrema of random number of variables

Keywords:
Distribution theory, Extremes, Marschall-Olkin generalized distribution, Quadratic transmutation map, Random minimum, Sibuya distribution, Stochastic representation

### Abstract

Recent years have seen an increased interest in the transmuted probability models,which arise from transforming a “base” distribution into its generalized counterpart. While manystandard probability distributions were generalized throughout this simple construction, the concept lacked deeper theoretical interpretation. This note demonstrates that the scheme is more

than just a simple trick to obtain a new cumulative distribution function. We show that the transmuted distributions can be viewed as the distribution of maxima (or minima) of a random number N of independent and identically distributed variables with the base distribution, where N has a Bernoulli distribution shifted up by one. Consequently, the transmuted models are, in fact, only a special case of extremal distributions defined through a more general N .