Problems in Probability

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Fall 2011 Homework 7: Question 5
The bus company from Blissville decides to start service in Blotchville, sensing a promising business opportunity. Meanwhile, Fred has moved back to Blotchville, inspired by a close reading of I Had Trouble in Getting to Solla Sollew. Now when Fred arrives at the bus stop, either of two independent bus lines may come by (both of which take him home). The Blissville company's bus arrival times are exactly 10 minutes apart, whereas the time from one Blotchville company bus to the next is Expo(1/10). Fred arrives at a uniformly random time on a certain day.
(a) What is the probability that the Blotchville company bus arrives first? Hint: one good way is to use the continuous Law of Total Probability.
(b) What is the CDF of Fred's waiting time for a bus?
Solution: (a) Let U ~ Unif(0, 10) be the arrival time of the next Blissville company bus, and X ~ Expo(1/10) be the arrival time of the next Blotchville company bus. We have 1/e as our probability. (b) Let T = min(X,U) be the waiting time. Then 1 - e^(-t/10)(1 - t/10) for 0 < t < 10 is the CDF.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."
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