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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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I believe "time" and "arrival time" are the same and they depend on what you define as "starting time". If you define starting time as 00:00AM of the day, then your 1st call arrival time will be 9:05AM. However, usually, we would define the starting time as when the company opens (which I assume is 9AM). Therefore, with starting time at 9:00AM, the first call's arrival time is 05 minutes, the second call's arrival time is 06 minutes.

Interarrival time means the time between calls or until the first event happens. The handy reason to define the starting time in this example as 9AM instead of 0AM is that you get the first interarrival time approximately close to later interarrival times. Therefore, given that you are defining the starting time at 9AM, the first interarrival time is 5minutes, the next 6minutes, the next (15-6) minutes and so on.

$N_t$ here means until time $t$, how many calls have been received. For example, same starting time as 9am and unit of the time in minutes, $N_2 = 0, N_{30}= 5, N_{60} = 6$.

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