Average passanger waiting time - probability density function - normal distribution

Hello guys, so we got this additional task to solve for a post-modul at university:

- The intermediate arrival time of a bus is normal distributed
- The expected value of the intermediate arrival time during rush hour is 4 minutes
- The standard deviation of the intermediate arrival time is 50 sec (50/60min)

A passanger enters the bus stop at a random time (statistically evenly distributed).

  1. What will be his waiting time for the next bus. How does the distribution (probability density function) for the passangers waiting time look like?
  2. What is the expected average waiting time of the passanger?
  3. Define the standard deviation of the passangers waiting time?
  4. Define the probability that the passanger has to wait more than 5 Minutes during rush hour?

I now how to solve it with a fixed arrival time of 4 minutes and how to get the standard deviation for that, but not how to solve it with a normal distribution and how to bring in the standard deviation.  Our professor mentioned something about forward recurrence time, but I couldn't find anything about that. Hope someone can help me out here.
Edit: An R Code would be fine too.


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