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Is it possible that the first arrival time of a poisson process is infinite?

A homogeneous Poisson process with intensity $\lambda > 0$ almost surely has a finite first arrival time, since $$\lim_{t \to \infty} \Pr[T \le t] = \lim_{t \to \infty} 1 - e^{-\lambda t} = 1.$$ ...
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$M$ Poisson r.v. with parameter $\lambda T(F(b)-F(a))$

Take $N_T=\max\Big\{n:\sum_{j=1}^n\tau_j\leq T\Big\}\sim \text{Poisson}(\lambda T)$. In the context of your problem, $N_T$ counts the numbers of points from $\Big\{\left(\sum_{j=1}^n\tau_j,X_n\right)\...
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Poisson Process with Changing Rate

Here is how you can compute $\mathbb{E}(T)$ without computing each $\mathbb{E}(T_i)$. For $t>0$ let $N_t$ represent the total number of firings on $[0,t]$. Take $X_t$ as the number of times counter ...
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2 votes
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Poisson Process with Changing Rate

Addendum: this answer computes an expression for $E[T_i]$ that is quite complicated. However, it is possible to compute $E[T]$ directly without computing each $E[T_i]$; see Matthew H's answer. Your ...
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