# Meaning of intersection of Poisson processes

Given a Poisson process $\left\{N_t\right\}_{t\ge \ 0}$, we need to find the following $P\left\{\cap \ _{r=0}^mN_r=2r\right\}$. But the aforementioned notation doesn't make any sense at all. What is the meaning of the intersection operator in this context? Please note I am not asking for the the answer to the expression itself and please note that I am aware of all the tools used in solving probabilities expectations related to Poisson processes taught in a typical introductory Stochastic processes class.

• I guess it is intended as the probability of the event $$\cap_{r=0}^{m} \{ N_r = 2r \} = \cap_{r=0}^{m} \{ \omega : N_r(\omega) = 2r \}.$$ It is the event that $N_r = 2r$ holds for all $r = 0, \cdots, m$. – Sangchul Lee Sep 28 '17 at 8:29
• Did you mean the following: $\left\{N_1=2\ and\ N_2=4\ and\ N_3=6\ ....\ and\ N_m=2m\right\}$ – fahad aijaz Sep 28 '17 at 11:34
• Yes, that is what I meant. – Sangchul Lee Sep 28 '17 at 12:10
• Thanks to you, the problem now seems trivial. – fahad aijaz Sep 28 '17 at 12:52