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.
Highlights
Students learn that the quadratic function
We focus on the problem of minimizing the function defined by y ( x) = E ( X – x ) in the context of mixtures of probability distributions of the discrete, absolutely continuous, and singular continuous types
The advantage of using Riemann-Stieltjes integration is that it covers the inequality for any discrete, continuous, and mixed type cdfs without exception with a single argument, which is presented in Theorem 1
Summary
The much less routine situation in which there is a discrete component and a continuous component in a single probability distribution is addressed An example of a mixed-type distribution is the lifetimes of electronic components. A third example of a mixed-type distribution is lifetimes in an experiment that is terminated at a predetermined time td. One choice is to standardize the continuous portion, so that it has a probability density This is usually expressed with conditional distributions The discrete portion might be standardized separately or ignored Another choice, which is to proceed with a mixed-type probability distribution for the whole experiment, is focused on presently. Continuous, and mixed-type distributions, including continuous distributions without a probability density function, which are discussed, are covered simultaneously with Riemann-Stieltjes integration Continuous, and mixed-type distributions, including continuous distributions without a probability density function, which are discussed in Section 3, are covered simultaneously with Riemann-Stieltjes integration ([1], p. 118-126), ([2], p. 11-14, 34), ([6], p. 281-284)
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