Waiting for the Bus: Schedule vs Random

If buses come exactly every 10 minutes, your wait is uniform between 0 and 10, averaging 5. If they arrive at random with the same 10-minute average gap, your expected wait is the full 10 minutes. For a continuous wait, probabilities are areas under the density curve.

How it works

Continuous distributions: uniform and exponential densities

In one sentence

A continuous random variable has a density $f(x) \ge 0$ with $\int f(x)\,dx = 1$ and $P(a \le X \le b) = \int_a^b f(x)\,dx$ . The uniform density on $[0,h]$ is $1/h$ , with mean $h/2$ ; the exponential density $\frac{1}{h}e^{-x/h}$ for $x \ge 0$ has mean $h$ and $P(X > a) = e^{-a/h}$ .

General formula

{\htmlData{sym=0}{f_U(x)}} = \frac{1}{\htmlData{sym=2}{{h}}}\;\;(0 \le x \le {\htmlData{sym=2}{h}}), \qquad {\htmlData{sym=1}{f_E(x)}} = \frac{1}{\htmlData{sym=2}{{h}}}e^{-x/{\htmlData{sym=2}{h}}}, \qquad P(X > {\htmlData{sym=3}{a}}) = \int_{\htmlData{sym=3}{a}}^{\infty} f(x)\,dx

For a continuous wait, the chance of any exact time is zero; only ranges of time have probability, measured as area.

What the symbols mean

$f_U(x)$
Scheduled-bus density
The flat density of your wait when buses come exactly every h minutes.
$f_E(x)$
Random-arrival density
The exponential density of your wait when buses arrive at random.
$h$
Average gap
The average number of minutes between buses.
$a$
Wait threshold
A wait you'd call long, in minutes.

A continuous wait has a density curve $f(x)$ : probabilities are areas under it, and the total area is 1.
Scheduled buses every $h$ minutes make your wait uniform on $[0, h]$ : a flat density of height $1/h$ , with average $h/2$ .
Buses arriving at random (a Poisson process) make the wait exponential, $f_E(x) = \frac{1}{h}e^{-x/h}$ , with average $h$ . You're more likely to show up during one of the long gaps.
The exponential is memoryless: after waiting 5 minutes, your remaining wait is still exponential with the same average.

Learn the math

Continuous Random VariablesStats · Lesson 18, Continuous Distributions and the Normal Curve The Uniform DistributionStats · Lesson 19, Continuous Distributions and the Normal Curve The Exponential DistributionStats · Lesson 20, Continuous Distributions and the Normal Curve Time to Failure: The Exponential ModelEng Stats · Lesson 5, Probability Models for Engineering

Learn the math

Continuous Random VariablesStats · Lesson 18, Continuous Distributions and the Normal Curve

The Uniform DistributionStats · Lesson 19, Continuous Distributions and the Normal Curve

The Exponential DistributionStats · Lesson 20, Continuous Distributions and the Normal Curve

Time to Failure: The Exponential ModelEng Stats · Lesson 5, Probability Models for Engineering