The following exercise is in my book:
The arrival process of ships into the bay is Poisson at a rate
$\lambda$. The probability that any ship is foreign is $p$, and it is
independent of other ships. Describe the arrival process of foreign
ships into the bay.
This is a difficult question for me because unlike other problems, there is no calculation in this one. I guess it is a good idea to understand this problem because it is testing your understanding of Poisson process.
I am not so sure what you can say about the arrival of foreign ships. I know the interarrival times of all the ships will be exponential with parameter $\lambda$. But what can I say about the arrival process of foreign ships? I guess there must be some reason why they gave me probability $p$. Maybe I can say something with that?
Can someone please help me?
Best Answer
It will be a Poisson process with rate $\lambda p.$ Roughly speaking, the probability of an arrival of a foreign ship in short time interval $dt$ will be the probability $\lambda dt$ that a ship arrives times the probability $p$ that the ship is foreign. Whether there is an arrival in the next small time interval will still be independent, so it is still a Poisson process.