Solution
Rates from Unequal Watching Times
Show the problem again
Events occur as a Poisson process with a constant unknown rate λ per hour. The process is watched during three separate, non-overlapping windows. In a 2-hour window 3 events are seen, in a 5-hour window 10 events are seen, and in a 1-hour window 0 events are seen. What is the maximum likelihood estimate of λ, in events per hour? Give an exact fraction.
Worked solution
The counts are independent Poisson variables with means 2λ, 5λ and λ. The log-likelihood is (13) ln λ − 8λ plus terms free of λ, so λ̂ = (total events)/(total time) = 13/8 = 1.625. Averaging the three per-window rates, (3/2 + 2 + 0)/3 = 7/6, gives a different (and inferior) answer because it gives the 1-hour window the same say as the 5-hour window.
Source: Original problem written for AxiomIQ.