(Solution) - Suppose that a certain large population contains k different types -(2025 Original AI-Free Solution)

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Suppose that a certain large population contains k different types of individuals (k ? 2), and let ?i denote the proportion of individuals of type i, for i = 1, . . . , k. Here, 0 ? ?i ? 1 and ?1 + . . . + ?k = 1. Suppose also that in a random sample of n individuals from this population, exactly ni individuals are of type i, where n1 + . . . + nk = n. Find the M.L.E.?s of ?1, . . . , ?k.