Calculating Poisson Process probabilities

[shan]

I just want to check my answers/reasoning as I'm not sure if I assumed the right things to do these problems.

N = {N(t), t>=0} ~ Poisson(1) and $$N_{(t,t+h]} = N(t+h)-N(t)$$

Determine P(N(4) =3|N(2) = 1)
Here I presumed that since N(2) = 1, then there must be 2 more arrivals in the interval (2,4] so that N(4)=3 so I calculated
$$P(N_{(0,2]} = 1 and N_{(2,4]} = 2) = P(N(2) = 1)P(N(2) = 2) = 4e^{-4}$$
since non-overlapping intervals are independent so I can multiply the probabilities (?). However I'm not too sure if I can break the intervals up like that?

Determine $$P(N_{(4,7]} = 2 and N_{(3,6]} = 1)$$
Here I split up the intervals into (3,4], (4,6] and (6,7] although like I said before, I'm not sure if that should be done. Then I looked for the different combinations of events over those intervals so that the above would be true ie
$$P(N_{(3,4]} = 0 and N_{(4,6]} = 1 and N_{(6,7]} = 1) + P(N_{(3,4]} = 1 and N_{(4,6]} = 0 and N_{(6,7]} = 2) = \frac{5e^{-4}}{2}$$
Again I assumed non-overlapping intervals were independent so I could multiply the probabilities together. It would be great if someone could double-check for me that those two are the only combinations (I did it three times already but I'm paranoid).

Determine $$P(N_{(4,7]} = 2|N_{(1,5]} = 2)$$
I used the same sort of steps as I did before although it's a conditional but I thought that if I split up the intervals and made it independent, conditionals would not matter. I used intervals (1,4], (4,5] and (5,6] and found the combinations
$$P(N_{(1,4]} = 2 and N_{(4,5]} = 0 and N_{(5,7]} = 2) + P(N_{(1,4]} = 1 and N_{(4,5]} = 1 and N_{(5,7]} = 1) + P(N_{(1,4]} = 0 and N_{(4,5]} = 2 and N_{(5,7]} = 0) = \frac{31e^{-6}}{2}$$

 

[Almsco]

Hi there! It seems like you have a good understanding of the Poisson process and your approach to the questions is generally sound. However, for the second and third questions you need to consider the fact that the intervals are not independent, since they overlap in some parts. Therefore, you can't simply multiply the probabilities together as you have done. I recommend looking into conditional probability and the law of total probability to help find the correct answer.