The Uniform distribution is the simplest probability distribution. Still, 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 have a rectangular distribution in 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 of the form $X \sim U(-2, 0.8)$. This is 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 by the equation

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

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Citation

The E-Learning project SOGA-R was developed at the Department of Earth Sciences by Kai Hartmann, Joachim Krois and Annette Rudolph. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: Hartmann, K., Krois, J., Rudolph, A. (2023): Statistics and Geodata Analysis using R (SOGA-R). Department of Earth Sciences, Freie Universitaet Berlin.