Probability of an event in Merged Poisson Process

Question

I have two Poisson processes A and B having arrival rates $$\lambda_{1}$$ and $$\lambda_{2}$$.

What is the probability that only 1 arrival appears from Process A and 0 arrival occurs from process B in the time interval [0-10s]? I mean what is the probability that in 10s time only 1 arrival from Process A appears?

Answer 1

Well, $~\mathsf P(A_{(0;10]}{=}1 \cap B_{(0;10]}{=}0) ~=~ \mathsf P(A_{(0;10]}{=}1)\,\mathsf P(B_{(0;10]}{=}0)~$ if they are independent processes, which you neglected to mention if it is so or not.

If so, it is just a case of applying the Poisson Distribution's probability mass function for the interval and rates.

Answer 2

here mean number of events per interval = m Process 1 mean arrival rate = m1 process 2 mean arriva rate = m2

P(A)= probability that there is an arrival from process1 P(B)= probability that there is no arrival from proecss 2 p(A|B)= P(A) as p(A^B) = P(A)P(B)/P(B)= P(A) thus, P(X=1)=(E^(-1)*1)/1= 1/e