- #1
FireSail
- 1
- 0
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 [tex]\lambda[/tex]1 and [tex]\lambda[/tex]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.
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 [tex]\lambda[/tex]1 and [tex]\lambda[/tex]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.