Final answer:
The student's question involves the application of the exponential distribution to model various scenarios concerning arrival and service times. The MGF, moments, probabilities, and percentiles can all be calculated using standard formulas for this distribution.
Step-by-step explanation:
The scenario described involves a probability distribution, likely exponential, that applies to the waiting times between customer arrivals and the duration of time spent with a clerk or finding a parking space. The exponential distribution is often used to model the time between events in a Poisson process, which is the key to understanding various probability queries in this context.
To calculate the moment generating function (MGF), one would typically integrate the product of the exponential function etx and the given probability density function (pdf) over the possible values of the random variable X, which represents time in this case.
The first moment or the expected value (E[X]), and the second moment (E[X2]) could then be obtained by differentiating the MGF. For an exponential distribution with a mean (λ-1), the first moment would be λ-1 and the second moment would be 2λ-2.
Given the scenarios, one would use the equations related to the exponential distribution to find the probabilities and percentiles asked for in the various parts of this question. For example, the probability that a customer arrives within a certain time frame can be computed by integrating the given pdf over that time frame or by using the cumulative distribution function.
For Example 5.7, where the amount of time spent by a postal clerk with a customer follows an exponential distribution with a mean of four minutes, we can find specific probabilities by using the property of the exponential distribution which says P(X > x) = e-x/λ for x ≥ 0. Therefore, to find the probability that a clerk spends four to five minutes with a customer, we’d evaluate P(4 < X < 5), which is the difference between P(X > 4) and P(X > 5).