Solution
Poisson Intervals with No Events
Show the problem again
Events arrive as a Poisson process with unknown rate λ per minute. A logger is too crude to count events: for each of 50 independent one-minute intervals it records only whether the interval contained no events. Exactly 20 of the 50 intervals were empty. What is the maximum likelihood estimate of λ? Give an exact closed form (a natural logarithm), or a decimal rounded to 3 or 4 places.
Worked solution
The data are 50 Bernoulli trials with success probability q = P(no events) = e^(−λ), and 20 successes. The MLE of q is the sample proportion 20/50 = 2/5. The likelihood depends on λ only through q, and q = e^(−λ) is a one-to-one map, so the MLE is carried along (invariance): e^(−λ̂) = 2/5, giving λ̂ = ln(5/2) ≈ 0.916. The trap is to guess from a mean count that you never observed; the only information is the fraction of empty intervals.
Source: Original problem written for AxiomIQ.