Polynomial Distributions and Transformations

Polynomials are common algebraic structures, which often used to approximate functions, such as probability distributions. This paper proposes directly define polynomial distributions in order describe stochastic properties of systems rather than assume polynomials for only approximating known or empirically estimated Polynomial offer great modeling flexibility and mathematical tractability. However, unlike canonical distributions, functions may have non-negative values the intervals support some parameter values; their numbers usually much larger interval must be finite. Hence, defined here assuming three forms a function. Transformations approximations histograms by also considered. The key derived closed form. A piecewise distribution construction is devised ensure that it over interval. goodness-of-fit measure proposed determine best polynomial. Numerical examples include estimation parameters generating polynomially distributed samples.