Two independent Poisson processes (one discrete, one continuous)

[FireSail]

Hi Guys,
I've used this forum as a great resource for a while now and it's always helped me out. Now I'm really stuck on something and was hoping you guys could help out. It's a pretty long question, but if you guys can just give me a general direction of what to do, I can go ahead and work it out for myself.

-----

Consider two independent Poisson processes N₁ and N₂ with rate parameters \lambda₁ and \lambda₂, respectively:

1. Find the prob. mass function for the number of events in N₂ that occur before the first event after time 0 of N₁ and identify what type of distribution it is.

So far my intuition to is to create a third process, N₃ = N₁ + N₂, and then calculate the probability P(N₃<0) from the joint distribution of N₁ and N₂. But I'm not sure this is the right way to do it.

The second part is trickier: Find the conditional density of the time of the first event after time 0 of N₁ given that there are x events in N₂ that occur before this first event of N₁. Also, for a given x, how should you predict the time of the first event of N₁ to minimize the mean squared error of your prediction?

The biggest problem I see is that I'm not sure how you're supposed to come up with a conditional probability if one random variable is discrete and the other is continuous. Can anyone point me in the right direction with this?

Thanks a ton you guys.

 

[GeorgeRaetz]

Might try this:
After time t, Probability for one event in process1 to occur over interval dt and N events in process 2 occurring after time t has elapsed should be the product:
λ₁ exp(-λ₁ t) exp(-λ₂ t) (λ₂ t)^N/N! dt
Integrate over all positive t to get answer to question 1.