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A bus will depart every 10 minutes from the origin, and the time it takes to travel to station $A$ follows a Poisson distribution with expectation of 10 minutes.

  1. Alice arrives at station $A$ around 9:00 AM, and her arrival time follows a uniform distribution with $a = -10, b = 10$. What is Alice's expected wait time station $A$?

  2. Bob arrives at station $A$ at 10:00 AM on the dot, what is Bob's expected wait time?

After reading this post: Please explain the waiting paradox

I think the answer to the first question is 5, since we can swap the role of the bus and Alice in the problem in Glen_b's answer, and arrive at the midpoint of the 10-minute intervals.

But for the second problem, what we are calculating is how long after 10:00 AM does the bus arrive, can we say it's 10 minutes by somehow shift the distribution, since 10:00 AM is a constant?

user101998
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    There's something seriously weird about this question: "the *time* it takes to travel to station A follows a *Poisson distribution* with expectation of 10 minutes" ... they're using a Poisson distribution for elapsed time. Times are continuous, the Poisson is discrete. In a Poisson process, it's event counts that are Poisson and the inter-event times are exponential. If the question really means what it says, be very careful about applying the information from the link, which does not use this strange formulation. ... – Glen_b May 14 '22 at 03:58
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    The properties of the Poisson don't make sense for times because the units don't match up. Standard deviation of times should be in minutes but the Poisson has variance = mean. Change the units to seconds or hours and you lose variance=mean, so it can't be Poisson. $:$ Indeed I am very concerned that whoever wrote the question potentially misunderstands the models being used so badly that they might expect almost any answer at all, depending on what else they misunderstand at the same time. – Glen_b May 14 '22 at 03:59
  • @Glen_b Thanks; I didn't expect to see you here! For the objection regarding a discrepancy of units, wouldn't it apply to all instances of Poisson distribution? For example, if the number of policies an insurance agent sells per week follows a Poisson distribution with mean of 3, the standard deviation would be $\sqrt{3}$ policies, and variance would be 3 policy squared? If this is indeed correct, would it help if we discretize time into minutes? Although it doesn't make sense to talk about a fractional policy from the insurance agent, we indeed got a non-integral for stand deviation. – user101998 May 14 '22 at 17:49
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    Counts are unitless (yes, they're a count of something, but the count itself does not have units). There's no problem there making variance=mean. Times are not unitless. There's definitely a problem making variance=mean there -- certainly enough that if you decided to use a Poisson for times, you'd need to explain how that worked (e.g. an underlying quasi-Poisson model that just happened for some reason to have $\phi=1$ in this instance for these units; but that sort of explanation makes no sense here, since the time is not a scaled count either). – Glen_b May 14 '22 at 23:31
  • It doesn't matter whether you round/truncate times to integers, the unit problem persists – Glen_b May 14 '22 at 23:34
  • Sorry, I'm not following the issue presented if the Poisson distribution is used to model "time". Does the underlying variable being modeled must not have units associated with them? If they do have unites (like "minute", rounded to the nearest integer for example), why doesn't the model work? – user101998 May 15 '22 at 15:53
  • As a general principle, I presume you're happy to say that two people each working with their own unit (say one that uses metric and one that uses 'US customary units', or one that converts monetary amounts to US dollars and one that converts them to Baht) should not come to different conclusions on the same sets of results? – Glen_b May 16 '22 at 00:36
  • Actually given this discussion is already about 500 words and (given my current explanation was insufficient) looks likely to end up much longer again than that, you should post a question about that issue of units and suitable models specifically. You may wish to nudge me here if you do. – Glen_b May 16 '22 at 00:38
  • @Glen_b Thanks, I have posted another question: https://stats.stackexchange.com/questions/575413/confusion-on-units-for-the-poisson-distribution-when-it-is-used-to-model-variabl – user101998 May 16 '22 at 02:11
  • In "the time it takes to travel to station follows a Poisson distribution with expectation of 10 minutes," I wonder if there might have been a misprint, and this should have read "exponential distribution" instead. – EdM May 16 '22 at 02:26

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