# 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).

- What will be his waiting time for the next bus. How does the distribution (probability density function) for the passangers waiting time look like?
- What is the expected average waiting time of the passanger?
- Define the standard deviation of the passangers waiting time?
- 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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