Piecewise Probability Distribution Theory

TL;DR

This paper develops a comprehensive theoretical framework for piecewise probability distributions, providing explicit formulas and algorithms for normalization, expectation, variance, median, and mode, applicable to various specific cases.

Contribution

It introduces a unified approach with explicit formulas and algorithms for one-dimensional piecewise linear probability distributions and their particular cases.

Findings

Derived explicit formulas for normalization, expectation, variance, median, and mode.

Validated formulas using known distributions like triangular and tetragonal distributions.

Provided algorithms applicable to general and specific piecewise linear distributions.

Abstract

A general piecewise (including pointwise) probability distribution with space-saving notation and its hierarchical particular cases are considered. The explicit closed-form normalization, expectation, and variance formulas along with the median and mode formulas and algorithms for a general one-dimensional piecewise linear probability distribution are obtained. They are also applied to a general polygonal, or one-dimensional piecewise linear continuous, probability distribution and, in particular, to a tetragonal probability distribution. The known formulas for the last distribution and a triangular probability distribution as a further particular case are used to test the obtained formulas and algorithms.