1. If X is a continuous random variable, P(a ≤ X ≤ b) is the same as P(a < X < b). Since these are different events, how can they have the same probability?

2. Someone who rides the subway back and forth to work each weekday makes about 40 trips a month. On the exponential subway, how many times a month can this commuter expect a wait longer than half an hour?

3. Find the probability that a rider of the exponential subway waits more than 20 minutes for a train; that is, find the probability of a wait more than twice as long as the average.

4. On the exponential subway, what is the probability that a randomly arriving passenger has a wait of between 9 and 10 minutes? What is the corresponding probability on the uniform subway?

5. If our system is aiming for an average interarrival time of 10 minutes, we might set a tolerance of plus or minus 2 minutes and try to keep the interarrival times between 8 and 12 minutes. Under the exponential model, what fraction of interarrival times fall in this range? How about under the uniform model?

6. Most mathematical software includes routines for generating “pseudo-random” numbers (that is, numbers that behave randomly even though they are generated by arithmetic). That’s what we used to simulate the exponential waiting times for our subway system. But the website (www.fourmilab.ch/hotbits/) delivers random numbers based on the times between decay events in a sample of Krypton-85. As noted above, the waiting times between decay events have an exponential distribution, so we can see what nature’s random numbers look like. See the short sample in Figure 2.

Actually, this source builds its random numbers from random bits, that is, 0’s and 1’s that occur with equal probability. See if you can think of a way of turning a sequence of exponential waiting times into a random sequence of 0’s and 1’s.

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