Polynomial Distributions and Transformations

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

The initial period function of late-type binary stars and its variation

The variation of the period distribution function of late-type binaries is studied. It is shown that the Taurus--Auriga pre-main sequence population and the main sequence G dwarf sample do not stem from the same parent period distribution with better than 95 per cent confidence probability. The Lupus, Upper Scorpius A and Taurus--Auriga populations are shown to be compatible with being drawn from the same initial period function (IPF), which is inconsistent with the main sequence data. Two possible IPF forms are used to find parent distributions to various permutations of the available data which include Upper Scorpius B (UScB), Chameleon and Orion Nebula Cluster pre-main sequence samples. All the pre-main sequence samples studied here are consistent with the hypothesis that there exists a universal IPF which is modified through binary-star disruption if it forms in an embedded star cluster leading to a general decline of the observed period function with increasing period. The pre-main sequence data admit a log-normal IPF similar to that arrived at by Duquennoy & Mayor (1991) for main sequence stars, provided the binary fraction among pre-main sequence stars is significantly higher. But, for consistency with proto-stellar data, the possibly universal IPF ought to be flat in log-P or log-semi-major axis and must be similar to the K1 IPF form derived through inverse dynamical population synthesis, which has been shown to lead to the main sequence period function if most stars form in typical embedded clusters.Comment: 13 pages, 8 figures, LaTeX, accepted by A&A, minor change to reference lis