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Let $(N_t)_t$ be a Poisson process with parameter λ = 2. By $τ_k$ denote the time of the k-th arrival (k = 1, 2, . . .), and by $ρ_k = τ_k −τ_{k−1}$ - the interarrival time between the (k−1)th and kth arrival (k = 1, 2, . . .), with $τ_0 = 0$ (as in the construction of Poisson process).
Find the following:
(a) $E(N_3N_4)$
(b) $E(ρ_3ρ_4)$
(c) $E(τ_3τ_4)$

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Calls are received at a company call center according to a Poisson process at the rate of five calls per minute.
(a) Find the probability that no call occurs over a 30-second period.
(b) Find the probability that exactly four calls occur in the first minute, and six calls occur in the second minute.
(c) Find the probability that 25 calls are received in the first 5 minutes and six of those calls occur in the first minute.

The above are some of the typical problems related to Poisson Process. I need to understand the difference between time, inter-arrival time, and arrival time in this regard.

Say, we start our counting from 9:00 AM and count up to 10:00 AM.

enter image description here

Image-1: arrival process.

  1. 1st call comes at 09:05 AM
  2. 2nd call comes at 09:06 AM
  3. 3rd call comes at 09:15 AM
  4. 4th call comes at 09:17 AM
  5. 5th call comes at 09:20 AM
  6. 6th call comes at 09:45 AM

enter image description here Image-2: Poisson counting process

What would be the time, arrival time, and inter-arrival time in this example?

As far as I understand:

  • 1 minute, 9 minutes, 2 minutes, 3 minutes, 25 minutes are inter-arrival-times $ρ_k$.

Are [0,5] minutes, [0,6] minutes, [0,15] minutes, ... etc time or arrival time?

If $(N_t)_t$ is the Poisson process, what would be the values of $t$?

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    $\begingroup$ I don’t think there is a globally agreed definition upon time vs arrival time. One refers to absolute (!) time and the other refers to relative time. But, your inter arrival times are correct. For further clarification, you can share your study resources / textbooks and we can try to spot the terminology used together. $\endgroup$
    – gunes
    Apr 13 '19 at 11:39
  • $\begingroup$ @gunes, kindly see the edit if it helps. $\endgroup$
    – user366312
    Apr 13 '19 at 11:59
  • $\begingroup$ According to the definitions in your first quotation, the arrival times are 5, 6, 15, 17, 20, and 45 minutes, exactly as you write. $\endgroup$
    – whuber
    Apr 13 '19 at 13:34
  • $\begingroup$ @whuber, what is $time$ then? $\endgroup$
    – user366312
    Apr 13 '19 at 14:55
  • $\begingroup$ The times are 9:05, 9:06, ..., 9:45. The one (crucial) piece of information they do not provide is that the process began at 9:00. $\endgroup$
    – whuber
    Apr 13 '19 at 15:30
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Ok, this is just an educated guess, especially based on your first example. But, still you can solve any of the listed exercises w/o discriminating between time vs arrival time.

The interarrival times are time distances between two consecutive events, and you’re correct in your call center example.

For the other two definitons, I think one source can easily use the term time in place of arrival time or vice versa in its description, and I haven’t seen so far any strict preference in textbooks on one over another.

In your first example, it says time of the k-th arrival ($k>0$) is $\tau_k$, which is a bit closer to arrival time in meaning. So, it is the relative time with respect to what you accept for $t=0$; in here it is $\tau_0$ since it is the $t=0$ point. And the term time is the absolute (except the fact that time is probably never absolute :) ) time in this context.

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  • $\begingroup$ I don't think this correct, for the simple reason that it contradicts the definition of $\tau_0$ in the first quotation of the question. $\endgroup$
    – whuber
    Apr 13 '19 at 14:24
  • $\begingroup$ @whuber this is a bit tricky to explain for me, tried my best and edited. $\endgroup$
    – gunes
    Apr 13 '19 at 14:32
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    $\begingroup$ Consider this: the distinction is that the (absolute) times tell you nothing about when the process started. The arrival times are relative to when it started. That is why the latter are more meaningful than, and distinguished from, the former. $\endgroup$
    – whuber
    Apr 13 '19 at 14:34

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