## 2.11.2 Exponential PDFA random variable T is said to have an exponential pdf ifThe mean, variance, and s-transform are The exponential pdf arises in numerous contexts, including Poisson processes. It has a "no-memory" property similar to that of the geometric pmf. We demonstrate this in the following example. Example 4: Jitney RiderSuppose that a person walks to the side of the roadway to wait for a jitney, which will transport her to the next town. A jitney is a
form of usually unscheduled transportation service involving minibuses
or macro-taxicabs that travel back and forth between two towns or other centers. Jitney
service, although uncommon in Europe and North America, is popular
in many countries throughout the world. Suppose that, by analysis
of past data or other means, it is known that the time required until
the arrival of the next jitney is an exponentially distributed random
variable with mean 10 minutes. Now suppose that our potential jitney
rider has already waited 15.5 minutes and she wants to know the
conditional mean additional time that she will have to wait.Solution:Hence, the conditional pdf is identical to the original exponential pdf but shifted to the right 15.5 units (Figure 2.11). By inspection, then, the conditional mean additional time that she has to wait remains 10 minutes. Here, 11 sunk investment" in waiting reaps no rewards in terms of reducing the remaining expected time until jitney arrival. Exercise 2.15: Jitney Rider, Revisited Redo Example 4 assuming that
the pdf for the time until arrival of the next jitney is uniform
between 2 and 18 minutes. How does sunk investment affect waiting
time in this case? |