Question

The bus arrives every 15 minutes starting at 8:00am and leaves immediately. You arrive at the bus stop with a uniform distribution between 8:05am and 8:30am and can be described as . Given that the bus arrival time and the time that you arrive at the bus stop are independent, what is the PDF of your

fx(x) = {1/25, 0<x< 25 0, otherwise

wait time?

Answer 1

The waiting time here ranges from 0 to 15 as we are given here that the bus arrives every 15 minutes here. From 8:05 to 8:10, the waiting time ranges from 0 to 10 minutes with a probability 10/25 = 0.4

Also from time 8:15 to 8:30, the waiting time ranges from 0 to 15, therefore, the probability of this happening is 1 - 0.4 = 0.6

Therefore the PDF for waiting time here is given as:

$f(w) = \left\{\begin{matrix} \frac{0.8}{10} &0 < w < 10 \\ \frac{0.2}{5} & 10 < w < 15 \end{matrix}\right.$

f(w) = 0.08 0 <w < 10 0.04 10 < W< 15

This is the required PDF for waiting time here.