I have some question about calculating some probability of an event.
I have a Poisson process,$N(t)$ t>=0, where N(t) is the counting of the number of event up to time t. And then I also have the $t_n$ which means the time of the $n^{th}$ event. My question is I was reading that: P({N(t)=n}) = $P(t_{n+1}>t)- P(t_n>t)$.
I think yes the probability on the right is equal to the probability on the left because:
Let A = the event that the $(n+1)^{th}$ event happened after time t Let B = the event that the $n^{th}$ event happened after time t
Then yes B is subset of A, and if I calculate $P(A \cap B^c) = P(A)-P(B) $, then it means the probability that the $n^{th}$ event happened before time t, and the $n+1^{th}$ must happen after time t, which is exactly the event N(t)=n. Could someone comment is this correct?
If it is correct, my question is when can we calculate probability of something say in this case P({N(t)=n}) by using the probability in another "space", so the other "space" is in the space t ( the time space I think in this case)?
My question is how do I know if I can calculate probability of some events in some spaces that seems to be a discrete variable, and that probability can somehow be calculated based on probability of events of some continuous variable?
I guess my question is it seems N(t) is different from $t_n$, but somehow the probability of event happening in N(t) can be calculated by event in $t_n$.
May I know in general when is this possible? and what is the mathematical foundation behind it? is it measure theory? product-measure? or what is it? sorry I don't know or understand enough to maybe even ask properly.
We have the fundamental relationship $$ \{N(t)=n\} = \{T_n\leqslant N(t) < T_{n+1}\} $$ for all $t\geqslant 0$ and $n=1,2,\ldots$ because $$ N(t) = \sum_{n=1}^\infty \mathsf 1_{(0,t]}(T_n). $$ Note that this is true for any renewal process, not just Poisson processes (i.e. when the increments $T_{n+1}-T_n$ are i.i.d. positive random variables and not necessarily exponentially distributed). It follows then that $$ \mathbb P(N(t)=n) = \mathbb P(T_n\leqslant N(t) < T_{n+1}). $$ It also true that $\mathbb P(N(t) = n) = \mathbb P(T_{n+1}>t) - \mathbb P(T_n>t)$, as $\mathbb P(N(t) = n) = e^{-\lambda t}\frac{(\lambda t)^n}{n!}$ while $$ \mathbb P(T_{n+1}>t) - \mathbb P(T_n>t) = \sum_{k=0}^n e^{-\lambda t}\frac{(\lambda t)^k}{k!} - \sum_{k=0}^{n-1} e^{-\lambda t}\frac{(\lambda t)^k}{k!} = e^{-\lambda t}\frac{(\lambda t)^n}{n!}. $$