The Uniform distribution is the simplest probability distribution, but it plays an important role in statistics since it is very useful in modeling random variables. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. The continuous random variable \(X\) is said to be uniformly distributed, or having rectangular distribution on the interval \([a, b]\). We write \(X \sim U(a, b)\), if its probability density function equals \(f(x) = \frac{1}{b-a}, \; x \in [a,b]\) and \(0\) elsewhere (Lovric 2011).

\[f(x) = \begin{cases} \frac{1}{b-a}, & \text{when $a \le x \le b$} \\[2ex] 0, & \text{when $x < a$ or $x > b$} \end{cases}\]

The figure below shows a continuous uniform distribution \(X \sim U(-2, 0.8)\), thus a distribution where all values of \(x\) within the interval [-2,0.8] are \(\frac{1}{b-a} (=\frac{1}{0.8-(-2)} = 0.36)\), whereas all other values of \(x\) are 0.

The mean and the median are given by \[ \mu = \frac{a+b}{2}\text{.}\]

The cumulative density function is shown below and given be the equation

\[F(x) = \begin{cases} 0, & \text{for $x < a$} \\[2ex] \frac{x-a}{b-a}, & \text{for $x \in [a,b)$} \\[2ex] 1, & \text{for $x \ge b$} \end{cases}\]