Question: There are N lottery tickets with numbers \(1,2,...,N\). However, N is unknown. In an unobserved moment, Uncle Ezra takes a ticket at random and memorizes its number. He puts it back, and repeats this experiment n times.

1. Provide Uncle Ezra’s maximum-likelihood estimator T for N. Is the estimator unbiased? (Hint: If X is a random variable with values in N, then \(E\left( X \right)=\sum\limits_{k\ge 1}{P\left( X\ge k \right)}\) )
2. Calculate approximately for large N the normalized expected value \({{E}_{N}}\left( T \right)/N\). (Hint: Think about what you have learned in calculus about Riemann sums.)

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