Poisson arrival time
= Poisson arrival time
{c}
{title2=$T_n=\inf\{t\geq0:N_t\geq n\}$}
The time of the $n$-th event of a <Poisson process>. At positive rate $\lambda$, the <Poisson interarrival times> $T_j-T_{j-1}$, with $T_0=0$, are <independent and identically distributed random variables> with the rate-$\lambda$ <exponential distribution>. In particular, $T_n$ has <expected value> $n/\lambda$ and <variance> $n/\lambda^2$. The inverse-count identity is $\{T_n\leq t\}=\{N_t\geq n\}$.