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FireSail

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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 N**

_{1}and N_{2}with rate parameters [tex]\lambda[/tex]_{1}and [tex]\lambda[/tex]_{2}, respectively:**1. Find the prob. mass function for the number of events in N**

_{2}that occur before the first event after time 0 of N_{1}and identify what type of distribution it is.So far my intuition to is to create a third process,

*N*, and then calculate the probability P(N

_{3}= N_{1}+ N_{2}_{3}<0) from the joint distribution of N

_{1}and N

_{2}. 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 N**

_{1}given that there are*x*events in N_{2}that occur before this first event of N_{1}. Also, for a given*x*, how should you predict the time of the first event of N_{1}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.