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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.

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  • $\begingroup$ 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 $r = 0, \cdots, m$. $\endgroup$ – Sangchul Lee Sep 28 '17 at 8:29
  • $\begingroup$ Did you mean the following: $\left\{N_1=2\ and\ N_2=4\ and\ N_3=6\ ....\ and\ N_m=2m\right\}$ $\endgroup$ – fahad aijaz Sep 28 '17 at 11:34
  • $\begingroup$ Yes, that is what I meant. $\endgroup$ – Sangchul Lee Sep 28 '17 at 12:10
  • $\begingroup$ Thanks to you, the problem now seems trivial. $\endgroup$ – fahad aijaz Sep 28 '17 at 12:52

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