SOA Fundamentals Of Actuarial Mathematics (FAM) Practice Test

In binomial MLE, which constraint must hold for the number of trials m given observed data?

m must be less than the smallest observed number of successes m must be at least the largest observed number of successes m equals the number of observations m must be a prime number

Correct Answer:

m must be at least the largest observed number of successes

Explanation:

In a binomial model, the number of trials m sets the maximum possible number of successes in a single experiment. Since each observed count of successes k must satisfy 0 ≤ k ≤ m, the data can only come from a scenario where m is at least as large as the largest observed count. If m were smaller than that maximum, that observation would have zero probability under the model, which is impossible. So the likelihood is only positive when m ≥ max(k_i) across all observations. That’s why the correct constraint is that m must be at least the largest observed number of successes. The other options don’t fit: m could be larger than the largest observed count, not restricted to be smaller; it isn’t tied to the number of observations; and it need not be a prime number.

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