TITLE:
Probability Models with Discrete and Continuous Parts
AUTHORS:
James E. Marengo, David L. Farnsworth
KEYWORDS:
Mixed-Type Distribution Function, Riemann-Stieltjes Integration, Median of a Random Variable, Cantor Distribution
JOURNAL NAME:
Open Journal of Statistics,
Vol.12 No.1,
February
21,
2022
ABSTRACT: In
mathematical statistics courses, students learn that the quadratic function is minimized when x is the mean of
the random variable X, and that the
graphs of this function for any two distributions of X are simply translates of each other. We focus on the problem of
minimizing the function defined by in the context of mixtures of probability distributions of the discrete,
absolutely continuous, and singular continuous types. This problem is important,
for example, in Bayesian statistics, when one attempts to compute the decision
function, which minimizes the expected risk with respect to an absolute error
loss function. Although the literature considers this problem, it does so only
under restrictive conditions on the distribution of the random variable X, by, for example, assuming that the
corresponding cumulative distribution function is discrete or absolutely
continuous. By using Riemann-Stieltjes integration, we prove a theorem, which
solves this minimization problem under completely general conditions on the
distribution of X. We also illustrate
our result by presenting examples involving mixtures of distributions of the
discrete and absolutely continuous types, and for the Cantor distribution, in
which case the cumulative distribution function is singular continuous. Finally,
we prove a theorem that evaluates the function y(x) when X has the Cantor distribution.