# Find a matching model for the following problem (probability task)

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

• 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? Commented Oct 10, 2017 at 17:04
• @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/… Commented Oct 10, 2017 at 17:06
• 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... Commented Oct 10, 2017 at 17:09
• @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! :) Commented Oct 10, 2017 at 17:28

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.)