Consider an urn which contains slips of paper each with one of the numbers 1, 2, . . . , 100 on it. Suppose there are i slips with the number i on it for i = 1, 2, . . . , 100. For example, there are 25 slips of paper with the number 25. Assume that the slips are identical except for the numbers. Suppose one slip is drawn at random. Let X be the number on the slip.

(a) Show that X has the pmf p(x) = x/5050, x = 1, 2, 3, . . . , 100, zero elsewhere.

(b) Compute P(X ≤ 50).

(c) Show that the cdf of X is F(x) = [x]([x] + 1)/10100, for 1 ≤ x ≤ 100, where [x] is the greatest integer in x.