Uniform distribution
High school
Consider a random experiment with outcomes in an interval $[a, b]$ (or in a finite set). The outcome has a uniform distribution if all outcomes are equally likely: for an interval, the probability of landing in a subinterval is proportional to its length,
\[P(c \leq X \leq d) = \frac{d - c}{b - a} \quad \text{for } a \leq c \leq d \leq b.\]Examples
Worked examples
-
Customers arriving at a shop independently of each other at an average rate of $3$ per hour
: If exactly one customer arrives between 1:00 and 2:00, then the arrival time: uniform on that hour
Undergraduate
For customers arriving independently at a constant rate, if exactly one customer arrives in a given hour, then the customer is equally likely to arrive at each time in that hour, so the arrival time has a uniform distribution on the hour.