Modeling With Continuous Uniform Distributions

Use the continuous uniform distribution to model real-world quantities and compute probabilities, means, and standard deviations in context.

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Tutorial

Setting Up a Continuous Uniform Model

A continuous uniform distribution models a quantity that is equally likely to take any value in an interval $[a,b].$ We write $X \sim U(a,b)$ to indicate that $X$ has this distribution.

The probability density function is

f(x) = \dfrac{1}{b-a}, \quad a \le x \le b,

and $f(x)=0$ outside $[a,b].$

The probability that $X$ falls in a sub-interval $[c,d] \subseteq [a,b]$ is the length of that sub-interval divided by the length of the support:

P(c \le X \le d) = \dfrac{d-c}{b-a}.

For instance, suppose a customer arrives at a store at a uniformly random time between 9:00 AM and 9:30 AM. Letting $X$ be the number of minutes past 9:00 AM, we have $X \sim U(0, 30),$ and

P(0 \le X \le 10) = \dfrac{10-0}{30-0} = \dfrac{1}{3}.