The simplest kind of continuous distribution is a rectangular distribution (also called a continuous uniform distribution).

Definition

A random variable, \(X\), is said to have a rectangular distribution with parameters \(a\) and \(b\)

\[ X \;\; \sim \; \; \RectDistn(a, b) \]

if its probability density function is

\[ f(x) = \begin{cases} \frac {\large 1} {\large b-a} & \text{for } a \lt x \lt b \\[0.2em] 0 & \text{otherwise} \end{cases} \]

Probabilities for rectangular random variables can be easily found using geometry.

Equivalently, using integration,

\[ \begin{align} P(c \lt X \lt d) \;\; &= \; \; \int_c^d {f(x)}\; dx \\ &= \; \; \int_c^d {\frac 1 {b-a}}\; dx \\ &=\;\; \frac {d-c} {b-a} \end{align} \]

Example

If \(X \;\; \sim \; \; \RectDistn(0, 10)\),

\[ P(4 \lt X \lt 7) \;\;=\;\; \frac {7-4} {10-0} \;\;=\;\; 0.3 \]