Uniform Distribution

The uniform distribution is a two-parameter continuous probability distribution. The table below summarizes some important aspects of the distribution.

Notation	\(X \sim \mathcal{U}(a, b)\)
Parameters	\(a \in (-\infty, b)\) (lower bound)
	\(b \in (a, \infty)\) (upper bound)
Support	\(\mathcal{D}_X = [a, b] \subset \mathbb{R}\)
PDF	\(f_X (x; a, b) = \begin{cases} \frac{1}{b - a} & x \in [a, b] \\ 0 & x \notin [a, b] \end{cases}\)
CDF	\(F_X (x; a, b) = \begin{cases} 0 & x < a \\ \frac{x - a}{b - a} & x \in [a, b] \\ 1 & x > b \end{cases}\)
ICDF	\(F^{-1}_X (x; a, b) = a + (b - a) \, x\)

The plots of probability density functions (PDFs), sample histogram (of \(5'000\) points), cumulative distribution functions (CDFs), and inverse cumulative distribution functions (ICDFs) for different parameter values are shown below.