Suppose that the Department of Transportation is doing a traffic study to determine the times that truckers visit a truck stop off of I-70, and they have a spotter at the exit to track the times that truckers enter. Let X(t) denote the number of truckers who are recorded by the spotter within the first t minutes of the study.
(a) [15 pts, 5 each] It would be nice if we could model X(t) as a Poisson process; however, that might not be warranted. For each of the following required properties of the Poisson process, give a common-sense reason why X(t) might fail to satisfy it in this situation:
i. Stationarity: X(t2) - X(t1) has the same distribution as X(s2) - X(s1) if t2 - t1 = s2 - s1. That is, the distribution of the number of occurrences of the process on a certain time interval only depends on the length of that interval, not "when" the interval is.
ii. Independent Increments: X(t2) - X(t1) is independent of X(s2) - X(s1) if the time intervals (t1, t2] and (s1, s2] do not overlap.
iii. Regularity: P(X(t+h) - X(t) > 1)/h -> 0 as h -> 0 – that is, occurrences can only happen one at a time.
Your answers above notwithstanding, let's assume that X(t) actually can be modeled as a Poisson process in a certain controlled situation, and the rate is λ = 2 truckers per minute.
(b) [10 pts] Find the exact probability that more than 250 truckers are counted during the first two hours of the study, and also provide a numerical answer to 4 decimal places (using Maple).
(c) [10 pts] Use the Central Limit Theorem to estimate the probability that more than 250 truckers are counted during the first two hours of the study, and report your answer to 4 decimal places. In your answer, indicate why the CLT can be used here!