• Sep 25th 2008, 04:40 AM
gloiterbox
Hi I have been stuck on this for days I do not think this is possible for me to do!!! I really need help. Thanks in advanced.

Ships pass a bird sanctuary at a Poisson rate of one per hour. Twenty percent of the ships are oil tankers.

1. What is the probability that at least one oil tanker will pass during a 24 hours.

2. If 30 ships have past by in one day, what is the probability that 6 of them were oil tankers.

Thanks again I am dying to know the answer!!!
• Sep 25th 2008, 05:02 AM
TKHunny
The joy of the Poisson process is that it is ENTIRELY scalable.

"Ships pass a bird sanctuary at a Poisson rate of one per hour."

This is a Poisson process with $\displaystyle \lambda\;=\;24$, referring to ships per day.

"Twenty percent of the ships are oil tankers."

24*0.2 = 4.8

This is a Poisson process with $\displaystyle \lambda\;=\;4.8$, referring to oil tankers per day.

"1. What is the probability that at least one oil tanker will pass during 24 hours."

1 - P(0) = ??

"2. If 30 ships have past by in one day, what is the probability that 6 of them were oil tankers."

This is a very odd question. It seems to have little to do with whether or not we have a Poisson process. I'm thinking a binomial model would be more appropriate.

p(oil tanker) = 0.2 = p
p(not oil tanker) = 0.8 = q = 1-p

$\displaystyle (p+q)^{30}$ = ???

What is the complete term with $\displaystyle p^{6}$ in it?

I suppose, since 30 is a relatively large number, that some other approximation might be appropriate. The binomial is simple enough in this case that playing with other defininitions may be off the mark.
• Sep 25th 2008, 05:16 AM
gloiterbox
Are you sure you can times the arrival rate like that? does it not have to be conditional on how many events have occured, or maybe smth to do with a compound poisson process. I thought of what you did first but then I came to the second part and it did not work there?
• Sep 25th 2008, 05:30 AM
TKHunny
I am sure that you cannot "times" anything, since that doesn't mean anything.

I stand by my original comments: "The joy of the Poisson process is that it is ENTIRELY scalable."

If you are forced to do a Poisson-type calculation, feel free to do so. Like I said, the Binomial seems most appropriate to me, but it is not the only way to proceed.

If there are 30 ships, then we expect 30*0.2 = 6 oil tankers. This could be a Poisson process with $\displaystyle \lambda = 6$, referring to tankers in every 30 ships. p(6) = ?? This produces a slightly lower value than the binomial version shown above, but it's in the neighborhood, as predicted. The problem with the Poisson Model is its right tail. It does not know that p(31) = 0.

The distributions are very similar:
• Sep 25th 2008, 08:24 AM
Laurent
As you probably know, the oil tankers pass at a Poisson rate of 1*0.2=0.2 per hour. (Take a Poisson process of rate $\displaystyle \lambda$, and select elements among the points of this process by picking each point with probability $\displaystyle p$, independently. What you end up with is a Poisson process of rate $\displaystyle p\lambda$). So the number of tankers that pass in 24 hours is a Poisson random variable of parameter 24*0.2=4.8. As a consequence, it is 0 with probability $\displaystyle e^{-4.8}$, and it is greater than or equal to 1 with probability $\displaystyle 1-e^{-4.8}\simeq 0.992$.

Among 30 ships, each of them is an oil tanker with probability 0.2, independently. So the number of oil tankers is a binomial random variable of parameters 30 and 0.2. In particular, the probability that exactly 6 ships are oil tankers is $\displaystyle {30\choose 6}0.2^6\cdot0.8^{24}\simeq0.179$. Notice that the fact that these ships passed in 24 hours is irrelevant to answer the question.