Uniform Distribution

A probability distribution in which every value within a defined range is equally likely to occur.

Source: Casella G, Berger RL. Statistical Inference. 2nd ed. Duxbury; 2002.

The uniform distribution assumes that every value within a defined range is equally likely, so no part of the range is favoured over another. In its continuous form it spreads probability evenly across an interval, as when any point between two limits is equally probable, while its discrete form gives each of a finite set of outcomes the same chance, like a fair die. It suits situations of pure ignorance or genuine equal likelihood, and it serves as a building block for simulation. Equal likelihood across a range is its defining feature. Kirkwood and Sterne (2003) describe this distribution.

Source: Kirkwood & Sterne 2003

The uniform distribution has a constant probability density over its range for the continuous form, or equal probabilities for each outcome in the discrete form, with a mean at the midpoint of the range and a variance determined by the width of the range. So the uniform distribution's properties include equal likelihood across its range, a mean at the centre, and a spread that increases with the range's width, and these simple properties, reflecting no preference among values within the bounds, are why it is used to represent situations of equal likelihood and as a basis for generating random numbers and other distributions.

Source: Casella & Berger 2002

The uniform distribution is used in simulation and random number generation, since uniform random values are the basis for generating draws from other distributions; as a model for situations where all outcomes in a range are equally likely; and, in Bayesian analysis, sometimes as a simple prior representing no preference among values. So the uniform distribution is used wherever equal likelihood across a range is appropriate or as a foundational tool, particularly in Monte Carlo simulation, where uniform random numbers are transformed to produce values from other distributions, which is why it is fundamental to computational statistics as well as a model of pure randomness within bounds.

Source: Casella & Berger 2002

The continuous uniform distribution assigns equal probability density to every value in a continuous range, so any value between the limits is equally likely, while the discrete uniform distribution assigns equal probability to each of a finite set of outcomes, such as the equal chances of a fair die's faces. So the two differ in whether the outcomes are continuous or discrete, with the continuous form having a flat density over an interval and the discrete form equal probabilities for each of a set of values, and both share the defining feature of equal likelihood, applied to a continuum or a finite set respectively.

Source: Casella & Berger 2002

The uniform distribution is appropriate when there is no reason to favour any value over another within a defined range, so all are treated as equally likely, or when modelling pure randomness within bounds, and it is used as a technical tool in simulation. So the uniform distribution is appropriate for representing complete uncertainty or equal likelihood across a range, such as a vague prior expressing no preference or a model of equally likely outcomes, and for generating random numbers, though it is a strong assumption of equal likelihood that is often unrealistic for real data, which is why it is used where that assumption genuinely applies or for computational purposes.

Source: Casella & Berger 2002