The expected waiting time for a bus that comes each c minutes

Question

This is not exactly a homework questions, but it's going to help me in solving another homework question (i,e. hints are great also ):

What's the expected waiting time for a passenger for a bus which arrived each $c$ minutes? The passenger will arrive at any time and assume the bus is operating 24/7.

Answer 1

Now that you have specified that the buses arrive regularly and (have sort of specified that) the passengers arrive randomly, the average waiting time is $\frac c2$ minutes. You can just focus on the interval before the next bus and it becomes clear. This is exactly what I meant with the comment that you need to ask a question that has a clear answer.

Answer 2

EDIT: This answer refers to the original version of OP's question.

I think there are a few information missing in this post. But the format of the question is way to familiar not to answer. It is basic queuing theory. I'll try.

The rate of arrival $c>0$ alludes to a Poisson distribution which makes sense in the context of arrivals, as it implies the exponentially distributed interarrival times which follow the Markov property.

Simple approach: Assume you are standing at the bus stop for a fixed, i.e. known unit of time. Assuming further that the random variable $N$ specifies the number of buses arriving with a rate of $c>0$ (the rate is specified in exactly the same unit of time) is Poisson-distributed, i.e. $N \sim Poi(c)$. Then, the probability of you 'catching' a bus (as several buses can arrive) is $$ \mathbb{P}(\mbox{'Catching a bus'}) = \mathbb{P}(N \geq 1) = \sum \limits_{k=1}^{\infty} \mathbb{P}(N=k) = \sum \limits_{k=0}^{\infty} \frac{c^{k}}{k!}e^{-c} = 1 - \frac{c^{0}}{0!} e^{-c} = 1 - e^{-c} \mbox{.}$$

In short, it is just the complementary event of no bus arriving during that time interval.

EDIT: This yields an expected waiting time of $\frac{1}{c}$ since interarrival times are exponentially distributed with the rate as mentioned before.

More elaborated approach: What is missing is definitely the (possibly probabilistic) nature of your arrival and the (possibly probabilistic) duration the bus actually stops for you to 'catch' it. This is way more complicated as we have to specify a joint probability measure on the 3 components. This would still neglect real life restrictions such as limitations on the number of buses able to stop at the same time etc.