How do I calculate the joint distribution for N events?

[Quaoar]

Hi,

I have a situation where there are N random events. These events can be from either one of two sources, one with a binomial distribution, the other with a Poisson distribution. The events do not provide any evidence of their origin, either distribution could be responsible for creating the event.

The question is, how do I calculate the probability that there will be N events given the parameters for both distributions?

 

[Hurkyl]

Your statement of the problem is a little unclear. But I suspect the answer is "add up the odds of all possible ways it could happen".

 

[Quaoar]

Your statement of the problem is a little unclear. But I suspect the answer is "add up the odds of all possible ways it could happen".

Well, basically each distribution pops out a random number of events, and these events are then observed as a total number of events. Looking at a particular event, I cannot tell if it originates from the binomial or the Poisson distribution.

So I guess what you're saying is, the probability of measuring say, 3 events, is:

P_1(0) * P_2(3) + P_1(1) * P_2(2) + P_1(2) * P_2(1) + P_1(3) * P_2(0)

Correct?

 

[Hurkyl]

Well, basically each distribution pops out a random number of events, and these events are then observed as a total number of events.

In other words, you're just looking at the sum of two independent random variables?

So I guess what you're saying is, the probability of measuring say, 3 events, is:

P_1(0) * P_2(3) + P_1(1) * P_2(2) + P_1(2) * P_2(1) + P_1(3) * P_2(0)

Sounds reasonable.