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. Given that the bus arrival time and the time that you arrive at the bus stop are independent, what is the PDF of your wait time? Graph the PDF of your wait time.

Answer 1

As the bus arrives every 15 minutes after 8am.

We are given here that the person arrives from 8:05 am to 8:30 am with uniformity here.

From 8:05 am to 8:15 am, the waiting time ranges from 0 to 10, as the bus arrives at 8:15. Also From 8:15 to 8:30, the waiting time ranges from 0 to 15, as the bus arrives at 8:30 here.

Therefore we have here:
P(0 < W < 10) = 10/25 = 0.4
P(0 < W < 15) = 15/25 = 0.6

Therefore P(0 < W < 10) = 0.4 + (2/3)*0.6 = 0.8

Therefore, P(10 < W < 15) = 1 - 0.8 = 0.2

Therefore the PDF for the 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 here.