# Merging Poisson Distributions

1. Aug 6, 2006

### DavidSmith

Consider a poisson process one (P1) with a frequency 'a' and if it happens 'k' times you get (e^-a)(a^k)/k!

and then you have another posssion processs that happens in the same time frame of P1 called P2 with a frequency of 'b' and if it happens 'z' times you get (e^-b)(b^z)/z!

So what is the probability that P1 merges with P2?

For example if is observed that a computer breaks once every 2 days and a lightbulb goes out every 4 days what is the probability that in one week two computers will break at the exact same time 5 lightbulbs go out?

I know this problem has something to do with the poisson distribution but I dont know how to merge these events to get an answer.

I went to wikipedia and searched on the net and couldnt find any exmaples of such a problem that deals with two events.

I found somethign on wikipedia that says:

If N and M are two independent random variables, both following a Poisson distribution with parameters λ and μ, respectively, then N + M follows a Poisson distribution with parameter λ + μ.

I know the the probabilities of both events happening are less than the probability of just one, but I dont think you can just multiply the probability of event each together to get the final probability of both events happening at the same time.

Last edited: Aug 6, 2006
2. Aug 6, 2006

### 0rthodontist

If they are independent Poisson processes, you just multiply the probabilities. The probability that five lightbulbs will go out over a period of time in which two computers break is the probability that five lightbulbs go out over that period times the probability that two computers break. I don't know what you mean by "exact same time," unless you mean one week.