A general integrable probability density function with an arbitrary number of open parameters

  • Harry H. Hilton University of Illinois at Urbana-Champaign
  • Ishan T. Karnik University of Illinois at Urbana-Champaign


A modified truncated Maclaurin series (MMS) formulation is presented in order to gain additional probability density function (PDF) parameters over the usual two or three associated with classical PDFs, such as Gauss, log-normal, Weibull, beta, etc. Such a protocol results in a higher fidelity match of experimental or other observational data due to the practically almost unlimited increase in number of available PDF parameters. Additionally, such a power series expression is readily analytically integrable in order to produce its associated cumulative distribution function (CDF), expectation and probability as well as all statistical moments. An illustrative example is included to exhibit a pattern of series convergence and errors of least square (LSQ) statistical data fits. Additionally, the relation between the Fischer sufficiency index, LSQ fit errors and number of series parameters is also investigated. Results have direct aerospace applications in failure probability analysis, statistical characterization of material properties, optimal design of systems of systems, random loads, sizing, thermal effects, etc.