Problems in Probability

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Fall 2011 Homework 6: Question 1
Fred lives in Blissville, where buses always arrive exactly on time, with the time between successive buses fixed at 10 minutes. Having lost his watch, he arrives at the bus stop at a random time (assume that buses run 24 hours a day, and that the time that Fred arrives is uniformly random on a particular day).
(a) What is the distribution of how long Fred has to wait for the next bus? What is the average time that Fred has to wait?
(b) Given that the bus has not yet arrived after 6 minutes, what is the probability that Fred will have to wait at least 3 more minutes?
(c) Fred moves to Blotchville, a city with inferior urban planning and where buses are much more erratic. Now, when any bus arrives, the time until the next bus arrives is an Exponential random variable with mean 10 minutes. Fred arrives at the bus stop at a random time, not knowing how long ago the previous bus came. What is the distribution of Fred's waiting time for the next bus? What is the average time that Fred has to wait? (Hint: don't forget the memoryless property.)
(d) When Fred complains to a friend how much worse transportation is in Blotchville, the friend says: "Stop whining so much! You arrive at a uniform instant between the previous bus arrival and the next bus arrival. The average length of that interval between buses is 10 minutes, but since you are equally likely to arrive at any time in that interval, your average waiting time is only 5 minutes." Fred disagrees, both from experience and from solving Part (c) while waiting for the bus. Explain what (if anything) is wrong with the friend's reasoning.
Solution: (a) The distribution is Uniform on [0, 10], so the mean is 5 minutes. (b) 1/4, conditioning on waiting time T > 6 thus far. (c) By the memoryless property, the distribution is Exponential with parameter 1/10 (and mean 10 minutes) regardless of when Fred arrives (how much longer the next bus will take to arrive is independent of how long ago the previous bus arrived). The average time that Fred has to wait is 10 minutes. (d) The average length of a time interval between 2 buses is 10 minutes, but this does not imply that Fred’s average waiting time is 5 minutes. This is because Fred is not equally likely to arrive at any of these intervals: Fred is more likely to arrive during a long interval between buses than to arrive during a short interval between buses. For example, if one interval between buses is 50 minutes and another interval is 5 minutes, then Fred is 10 times more likely to arrive during the 50 minute interval. This phenomenon is known as length-biasing, and it comes up in many real-life situations. For example, asking randomly chosen mothers how many children they have yields a different distribution from asking randomly chosen people how many siblings they have, including themselves. Asking students the sizes of their classes and averaging those results may give a much higher value than taking a list of classes and averaging the sizes of each (this is called the class size paradox).
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."
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