# Poisson process

• Oct 19th 2010, 01:17 AM
harbottle
Poisson process
Hi,

Suppose we have a poisson process with rate parameter = 2 per hour that counts the number of insects that jump into my soup. Each insect is green with probability 0.4, independently of the other colours.

Given that exactly one green insect jumped into my soup during a 45 minute dinner, what is the probability that at least one insect of another colour also jumped in?

Now, the "independently of other colours" bit confused me a little. Does that mean I can disregard the 1 green insect and just treat the differently-coloured insects as a separate poisson process with rate parameter 0.6*2 per hour? Surely not, since if a green insect jumped in, then the chance I'd have an insect of a different colour would be different, since there would need to be at least two insects in there, not at least one.

I've written down P(at least one differently coloured insect & exactly 1 green insect)/P(1 exactly one green insect) but as these two events are not independent, I cannot see where to go from here. Any help would be greatly appreciated.
• Oct 19th 2010, 11:31 PM
CaptainBlack
Quote:

Originally Posted by harbottle
Hi,

Suppose we have a poisson process with rate parameter = 2 per hour that counts the number of insects that jump into my soup. Each insect is green with probability 0.4, independently of the other colours.

Given that exactly one green insect jumped into my soup during a 45 minute dinner, what is the probability that at least one insect of another colour also jumped in?

Now, the "independently of other colours" bit confused me a little. Does that mean I can disregard the 1 green insect and just treat the differently-coloured insects as a separate poisson process with rate parameter 0.6*2 per hour? Surely not, since if a green insect jumped in, then the chance I'd have an insect of a different colour would be different, since there would need to be at least two insects in there, not at least one.

I've written down P(at least one differently coloured insect & exactly 1 green insect)/P(1 exactly one green insect) but as these two events are not independent, I cannot see where to go from here. Any help would be greatly appreciated.

The number of green-insects jumping into your soup is a Poisson RV with rate $\displaystyle 0.4 \times 2$ per hour

The number of non-green-insects jumping into your soup is a Poisson RV with rate $\displaystyle 0.6 \times 2$ per hour

(and these process are independednt)

CB