The uniform distribution describes a random variable that is equally likely to take any value in its sample space.

Overview

The uniform distribution describes a random variable that is equally likely to take any value in its sample space. A discrete random variable that is uniformly distributed on a set of outcome $\{1, 2, ..., k\}$ has probability mass function

$$P(X=x) = \frac{1}{k}$$

The continuous uniform distribution on $(a,b)$ has density

$$p(x)=\begin{cases} \frac{1}{b - a} & \mathrm{for}\ a \le x \le b, \\[8pt] 0 & \mathrm{for}\ x<a\ \mathrm{or}\ x>b \end{cases} $$

The uniform distribution is important in certain sampling methods, such as acceptance-rejection sampling and inverse transform sampling. In a Bayesian context, the uniform distribution can be used as an uninformative prior.

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