On the degree of uniformity measure for probability distributions

Abstract A key challenge in studying probability distributions is quantifying the inherent inequality within them. Certain parts of the distribution have higher probabilities than others, and our goal is to measure this inequality using the concept of mathematical diversity, a novel approach to examining inequality. We introduce a new measure m D ( P ), called the degree of uniformity measure on a given probability space that generalizes the idea of the slope of secant of the slope of diversity curve. This measure generalizes the idea of degree of uniformity of a contiguous part ( P = { k 1 , k 2 } in the discrete case or P = ( a , b ) in the continuous case) in a probability space related to a random variable X , to an arbitrary measurable part P . We also demonstrate the truly scale free and self-contained nature of the concept of degree of uniformity by relating the measure of two parts P 1 and P 2 from completely unrelated distributions with random variables X 1 and X 2 that have completely different scales of variation.