Rectangular distribution

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

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

In many applications of the rectangular distribution, the parameters \(a\) and \(b\) are known constants, but occasionally one of them is unknown.

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

We now show how integration can be used to get this probability.

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

Example

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

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