Fix a window of time $[0,T]$ and say that we get $n$ arrival times in the window from a homogeneous Poisson process. The maximum likelihood estimate (MLE) is just $n/T$ I believe. But what is the likelihood function for this Poisson process?
2026-04-12 16:55:40.1776012940
Likelihood function of a Poisson process
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If the rate is $r$ per unit of time then the parameter is $\lambda = rT$ so the likelihood function is $$(rT)^n \frac{e^{-rT}}{n!}$$
If you take the derivative of this with respect to $r$ and set this equal to $0$ to solve to find the maximum likelihood estimate of $r$, you do not get $T/n$. This is in fact obvious from dimensional analysis. But you do get something closely related, so perhaps you are thinking about some other parameter.