Model a waiting time

An exponential random variable models a nonnegative waiting time with a constant event rate. Its density starts at λ and decays; probabilities come from area, not the height at one time.

The survival probability P(T > t) equals exp(−λt), and both the mean and standard deviation equal 1/λ. A quantile is found by setting the cumulative probability and solving a logarithmic equation.

\displaystyle f(t)=\lambda e^{-\lambda t},\quad P(T>t)=e^{-\lambda t},\quad E(T)=1/\lambda

WORKED EXAMPLE

A waiting time has rate 0.25 per minute. Find its median.

At the median m, P(T > m) = 0.5.
Solve exp(−0.25m) = 0.5.
m = ln(2)/0.25 ≈ 2.773 minutes.

Assumed knowledge

Interpret a continuous probability density and find probabilities by integration. Use exponential functions, natural logarithms and inverse operations to solve equations.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Model a waiting time

A waiting time has rate 0.25 per minute. Find its median.

Put the idea to work.