In Self-Test Problem 1, we showed how to use the value of a uniform (0, 1) random variable (commonly called a random number) to obtain the value of a random variable whose mean is equal to the expected number of distinct names on a list. However, its use required that one choose a random position and then determine the number of times that the name in that position appears on the list. Another approach, which can be more efficient when there is a large amount of replication of names, is as follows: As before, start by choosing the random variable X as in Problem. Now identify the name in position X, and then go through the list, starting at the beginning, until that name appears. Let I equal 0 if you encounter that name before getting to position X, and let I equal 1 if your first encounter with the name is at position X. Show that E[mI] = d.
Problem 1
Consider a list of m names, where the same name may appear more than once on the list. Let n(i),i = 1,... ,m, denote the number of times that the name in position i appears on the list, and let d denote the number of distinct names on the list.
(a) Express d in terms of the variables m, n(i),i = 1,..., m. Let U be a uniform (0, 1) random variable, and let X = [mU] + 1.
(b) What is the probability mass function of X?
(c) Argue that E[m/n(X)] = d.
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