Polynomials are common algebraic structures, which are often used to
approximate functions including probability distributions. This paper proposes
to directly define polynomial distributions in order to describe stochastic
properties of systems rather than to assume polynomials for only approximating
known or empirically estimated distributions. Polynomial distributions offer a
great modeling flexibility, and often, also mathematical tractability. However,
unlike canonical distributions, polynomial functions may have non-negative
values in the interval of support for some parameter values, the number of
their parameters is usually much larger than for canonical distributions, and
the interval of support must be finite. In particular, polynomial distributions
are defined here assuming three forms of polynomial function. The
transformation of polynomial distributions and fitting a histogram to a
polynomial distribution are considered. The key properties of polynomial
distributions are derived in closed-form. A piecewise polynomial distribution
construction is devised to ensure that it is non-negative over the support
interval. Finally, the problems of estimating parameters of polynomial
distributions and generating polynomially distributed samples are also studied.Comment: 21 pages, no figure
