Suppose you walk to a bus stop to catch a bus; busses of the line you take arrive there every 15 minutes. You don’t look at your watch and arrive at the bus stop totally at random. For simplicity, we assume that all arriving times (you/the busses) are rounded down to whole minutes. When you and the bus arrive simultaneously, you will be able to catch it. a) Describe the probability space that models the above-described waiting time experiment, i.e. sample space and the related probabilities. Suppose now that you still start walking towards the bus stop at a completely random time; but that you do look at your watch at the beginning of your journey. Knowing your walking pace and the distance to the bus stop, you know by how many minutes you will miss the previous bus. If you would miss the bus by 4 or less minutes (i.e. hypothetically, the waiting time would be 11 minutes or more), you decide to hurry up; in this case, you will still be able to catch it. Otherwise, if you would miss the bus by 5 or more minutes (i.e. waiting times 10 minutes or less), you simply go on with your normal walking pace. b) Let the random variable X model your waiting time at the bus station in the just-described situation. Calculate the probability that you will need to wait for at least 5 minutes. c) Calculate the expectation of X. d) Calculate the variance of X. e) Suppose you walk to the bus stop in the previously described way year 160 times per year. Denote by X1, . . . , X160 your waiting times on all these days; assume they are independent of each other. Describe the (approximate) distribution of your average waiting time ̄X160 = 1/n * ∑^160_i=1 (Xi) across the whole year.

Suppose you walk to a bus stop to catch a bus; busses of the line you take arrive there every
15 minutes. You don’t look at your watch and arrive at the bus stop totally at random. For simplicity, we
assume that all arriving times (you/the busses) are rounded down to whole minutes. When you and the bus
arrive simultaneously, you will be able to catch it.
a) Describe the probability space that models the above-described waiting time experiment, i.e. sample space
and the related probabilities.
Suppose now that you still start walking towards the bus stop at a completely random time; but that you do
look at your watch at the beginning of your journey. Knowing your walking pace and the distance to the bus
stop, you know by how many minutes you will miss the previous bus. If you would miss the bus by 4 or less
minutes (i.e. hypothetically, the waiting time would be 11 minutes or more), you decide to hurry up; in this case,
you will still be able to catch it. Otherwise, if you would miss the bus by 5 or more minutes (i.e. waiting times
10 minutes or less), you simply go on with your normal walking pace.
b) Let the random variable X model your waiting time at the bus station in the just-described situation.
Calculate the probability that you will need to wait for at least 5 minutes.
c) Calculate the expectation of X.
d) Calculate the variance of X.
e) Suppose you walk to the bus stop in the previously described way year 160 times per year. Denote by
X1, . . . , X160 your waiting times on all these days; assume they are independent of each other. Describe
the (approximate) distribution of your average waiting time ̄X160 = 1/n * ∑^160_i=1 (Xi) across the whole year.