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Find a matching model for the following problem:

At a gas station, there are in average $3$ cars every minute, between 9:00 and 11:00. Let $Y$ be the random quantity of the arriving cars (cars that arrived between 9:00 and 11:00).

$$Y \sim$$

I was never confronted with such a task before. But I just experienced it might be asked in our exam :p

So I'm not quite sure what I'm supposed to do or rather how to create such a model by this text. I cannot create "my own" model but I know a model which already exists which might even represent that one explained in this text; it's the Poisson distribution. Because it's used to determine the amount of something in a specific time.

Is that correct? If so, how would you write it? Like that:

$$Y \sim Poi(\lambda), \text{ where } P(Y=k) = \frac{\lambda^k}{k!}e^{-\lambda}$$ ?

I hope everything is alright? Or is something completely different asked? :s

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    $\begingroup$ Yes Poisson process is used to model a stream of arrivals with constant mean rate in independent time intervals. Your notation to write down the model is fine. Was this really all you had to ask? $\endgroup$ Commented Oct 10, 2017 at 17:04
  • $\begingroup$ @LoveTooNap29 That's good to know. I asked this question because I was afraid I did it completely wrong. I was told that it will be asked in our exam and I didn't want fail at this part. Here is an example exam with some of these tasks but it's in German: www2.informatik.hu-berlin.de/~koessler/MathematikInformatiker/… $\endgroup$
    – cnmesr
    Commented Oct 10, 2017 at 17:06
  • $\begingroup$ Good luck on the exam. You mention being afraid that you "did it completely wrong", just out of curiosity, what made you think this? Because you were completely correct... $\endgroup$ Commented Oct 10, 2017 at 17:09
  • $\begingroup$ @LoveTooNap29 Hehe well I'm not sure how to explain it but in general I'm an uncertain person and I mostly need a confirmation, whatever it is. And thank you! :) $\endgroup$
    – cnmesr
    Commented Oct 10, 2017 at 17:28

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I agree with @LoveTooNap that a Poisson model is reasonable. However, you need to be careful about the rates. The question says the rate is $\lambda = 3$ cars per minute. Then asks you to model cars in a 2-hour period. So you have to adjust the Poisson rate for $Y$ to match the 2-hour period. (Also, notice that a Poisson random variable with a relatively large rate is nearly normal.)

The plot below may help. Black bars represent Poisson probabilities, the red curve is a normal density curve. (Ignore uneven shading; it is an artifact of printing on this page.)

enter image description here

Finally, unless there are enough pumps that on average somewhat more than three cars can be served in a minute, lines will form.

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