In front of the university campus in Diepenbeek, there is a bus stop at which there leaves every 20 minutes a bus to Hasselt. More, specifically, a bus leaves at the hour, 20 min past the hour or 40 min past the hour. When you do not worry about the time schedule and arrive at random at the bus stop, you can show that the average waiting time to the next bus is always 10 minutes by using a continuous uniform distribution. Just after the June exam period, the bus company has slightly changed the bus schedule. The buses now come at the hour, 10 min after the hour or 30 min after the hour. If you assume that you still not worry about the time schedule and arrive at random at the bus stop, how will the average waiting time for the next bus change? To solve this problem, we define the random variables: X = Time until the next bus when you arrive at random at the bus stop Y = the time point in the hour when you arrive at random at the bus stop a. What is the probability that you randomly arrive at the bus stop between 25 min past the hour and 40 min past the hour. b. Give the density of the random variable Y. e. Give the conditional density of the random variable X, given that Y = y. d. Are the variables X and Y independent of each other? e. Calculate the average waiting time to the next bus E[X].