Review on likelihood functions, ML estimates and estimator 1. There are at most 100 identical balls in a bowl but we don't know the exact number of them. However, we know that the balls are numbered with numbers 1, 2,...,0 and each of the number from number on to number is used exactly once and the > 1 is an integer. We pick a single ball in random. Let Y be the number in a ball we pick at random. (a) What is the parameter space in the statistical model corresponding this setup? (b) What is the probability mass function f(y; 0) of the random variable Y? (c) Suppose that the obseverd number is y = 57. Tell what is the likelihood function corresponding to this data. What is the maximum likelihood estimate of the parameter with this observation. What is the maximum likelihood estimator of the parameter 0? (d) Show by calculating that the espected value of the maximum likelihood estima- tor (Y) is E(Y) = 0+1 2 What is the bias of the ML estimator biase (⑦(Y))? (Hint: the English Wikipedia page https://en.wikipedia.org/wiki/Discrete_uniform_distribution might be useful)

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