= Solution
The <likelihood function> for the ordered arrival times of a <Poisson process>, including the absence of further arrivals before $T$, is
$$
\boxed{L(\lambda)=\lambda^n e^{-\lambda T}},\qquad 0<t_1<\cdots<t_n<T.
$$
The arrival-time constraint does not depend on $\lambda$. Equivalently, the <Poisson distribution> of the total count gives $L(\lambda)=e^{-\lambda T}(\lambda T)^n/n!$, which differs by a parameter-independent factor. Conditional on the count, the <Poisson process conditional arrival times> have density $n!/T^n$, independent of $\lambda$. Thus \b[the total count contains all the information about the rate when the exposure is known]; it is a <sufficient statistic>.
Back to article page
