MLE for the number of lottery balls

Question

A drum holds balls numbered 1, 2, \dots, N for unknown N, and each draw (with replacement) is uniform on \{1, \dots, N\}. Five draws produce 14, 71, 33, 68, 52. What is the maximum likelihood estimate of N?

Accepted Answer

The likelihood is L(N) = N^{-n}\,\mathbf{1}\{N \ge \maxi xi\}: zero unless N is at least the biggest value seen, and decreasing in N beyond that. So the maximizer is the sample maximum, \hat N = \maxi xi = 71. Takeaway. The MLE picks the smallest N still able to have produced every observed number, namely the largest one you saw.

MLE for the number of lottery balls

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