Uniform Distribution

The most basic of all the continuous random variables is the uniform random variable, which is equally likely to take on any value in its range (α,β). X is a uniform random variable (X∼Uni(α,β)) if it has PDF: f(x)={1β−αwhen α≤x≤β0otherwise

Notice how the density 1/(β−α) is exactly the same regardless of the value for x. That makes the density uniform. So why is the PDF 1/(β−α) and not 1? That is the constant that makes it such that the integral over all possible inputs evaluates to 1.

Uniform Random Variable

Notation: X∼Uni(α,β)
Description: A continuous random variable that takes on values, with equal likelihood, between α and β
Parameters: α∈R, the minimum value of the variable.
β∈R, β>α, the maximum value of the variable.
Support: x∈[α,β]
PDF equation: f(x)={1β−αfor x∈[α,β]0else
CDF equation: F(x)={x−αβ−αfor x∈[α,β]0for x<α1for x>β
Expectation: E[X]=12(α+β)
Variance: Var(X)=112(β−α)2
PDF graph:
Parameter α:
Parameter β:

Example: You are running to the bus stop. You don’t know exactly when the bus arrives. You believe all times between 2 and 2:30 are equally likely. You show up at 2:15pm. What is P(wait < 5 minutes)?

Let T be the time, in minutes after 2pm that the bus arrives. Because we think that all times are equally likely in this range, T∼Uni(α=0,β=30). The probability that you wait 5 minutes is equal to the probability that the bus shows up between 2:15 and 2:20. In other words P(15<T<20): P(Wait under 5 mins)=P(15<T<20)=∫1520fT(x)∂x=∫15201β−α∂x=130∂x=x30|1520=2030−1530=530

We can come up with a closed form for the probability that a uniform random variable X is in the range a to b, assuming that α≤a≤b≤β: P(a≤X≤b)=∫abf(x)dx=∫ab1β−αdx=b−aβ−α