# Two independent Poisson processes (one discrete, one continuous)

## Main Question or Discussion Point

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.

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Consider two independent Poisson processes N1 and N2 with rate parameters $$\lambda$$1 and $$\lambda$$2, 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.