Mr. Edwards is at a train station where local trains arrive once/$5$ mins, and express trains arrive once/$15$ mins in such a way that the every third local train arrives simutaneously with the express train. The local train takes $17$ mins travel time and the express train takes $11$ mins.
Let $T$ be the time Mr. Edwards takes to get to his destination including waiting time for the train and ride time.
(a) What's the waiting time distribution until the next local train arrives? The next express train?
(b) What's the probability that the next local train arrives alone without an express train. How about the probability that the next two local trains arrive alone?
(c) If the next local train arrives alone, is it optimal for Mr. Edwards to get on that train or keep waiting?
(d) Find $E(T)$ using your strategy described in (c).
(a) the waiting distribution for the next local train is $U(0, 5)$, and it is $U(0, 15)$ for the express train.
(b) I am not so sure about this. The back of the book has a hint which says "interpret probabilities as fractions of time," but that hasn't really gotten me anywhere. I guess $1/3$ because there is three buses, and the express bus will come with one of them. But I don't think this is right?
(c, d). For (c) I tried to make two different random variables and set another random variable to their minimum. I didn't get anywhere with this
Best Answer
Your answer to part a) is incorrect. The waiting time doesn't depend on the transit time, but on the arrival times. The distribution of the waiting time for a local is $U(0,5)$ and that of an express is $U(0,15)$.
As for part b), consider the $15$-minute interval starting just after one express leaves, and ending just after the next express arrives. If Mr. Edwards arrives any time in the first $10$ minutes of the interval, the next local will arrive alone. If he arrives in the last $5$ minutes, the next local will arrive at the same times as the express. Since the distribution of Mr. Edwards's arrival is uniform, the probability that the next local arrives alone is ${10\over15}=\frac23$.
The answer to part c) is "It depends." If the next local will arrive at the same time as the express, Mr. Edwards does better to wait $5$ minutes for the express, since he will then have a total transit time of $16$ minutes ($5$ minutes waiting and $11$ minutes traveling), which is better than riding the local for $17$ minutes. If the next local will arrive alone, then Mr. Edwards should take the train that's already there.
Now for part d). Now matter when he arrives, Mr. Edwards spends an average of $2.5$ minutes waiting. (This is the mean of $U(0,5)$.) If he arrives in the first $5$ minutes of the $15$-minute interval minutes between expresses, he will board the first local, and travel for $17$ minutes. If he arrives in the second $5$-minute interval, he will travel for $16$ minutes, as we have seen. If he arrives in the last $5$-minute interval, he will take the express and travel for $11$ minutes.
So there are $3$ possibilities. If he arrives in the first $5$ minutes, his travel time is $17$ minutes. If he arrives in the second $5$-minute interval, the transit time is $16$ minutes. If he arrives in the last interval, the travel time is $11$ minutes. The total expected time is $$2.5+{17+16+11\over3}=17.1666\dots$$
EDIT
In response to the OP's comment, the trains arrive on a fixed schedule, so I think it is only reasonable to think that Mr. Edwards knows how long it will be before the next express arrives. Assuming he does not know, however, then Mr. Edwards wants to minimize his expected travel time. The probability that the next local will arrive at the same time as an express is $\frac12$. If he waits for the next local, and it arrives without an express, he will end up waiting for the express, as we have seen. Therefore his expected transit time if he does not board the local is $$5+.5\cdot11+.5\cdot16=18.5$$ compared to the transit time of $17$ minutes if he boards the local, so he should just board the first train that comes, taking the express in preference to the local of course.
I think my original interpretation is probably the one the questioner intended, but there is no way to be certain of that, of course.