On a Probability Distribution

Abstract

In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. In technical terms, a probability distribution is a probability measure whose domain is the Borel algebra on the reals. A probability distribution is a special case of the more general notion of a probability measure, which is a function that assigns probabilities satisfying the Kolmogorov axioms to the measurable sets of a measurable space. Additionally, some authors define a distribution generally as the probability measure induced by a random variable X on its range - the probability of a set B is P(X − 1(B)). However, this article discusses only probability measures over the real numbers. Every random variable gives rise to a probability distribution, and this distribution contains most of the important information about the variable. If X is a random variable, the corresponding probability distribution assigns to the interval [a, b] the probability Pr[a ≤ X ≤ b], i.e. the probability that the variable X will take a value in the interval [a, b].