Solution
How Many Trials Were There?
Show the problem again
A number n of independent trials, each succeeding with probability 3/10, was performed, but the value of n was lost. All that is known is that there were exactly 5 successes. What is the maximum likelihood estimate of n?
Worked solution
Write L(n) = C(n, 5)(3/10)^5 (7/10)^(n−5). Then L(n)/L(n−1) = (n/(n−5))·(7/10), which is at least 1 exactly when 7n ≥ 10n − 50, i.e. n ≤ 50/3 = 16.67. So the likelihood rises up to n = 16 and falls afterwards, and the MLE is the integer 16 = ⌊5/(3/10)⌋. Rounding 5/0.3 to the nearest integer would give 17, which is wrong.
Source: Standard exercise on MLE for the binomial index n. Statement written for AxiomIQ.