Question

Let m ≥ 3. An urn contains m balls labeled 1, . . . , m. Draw all the balls from the urn one by one without replacement and observe the labels in the order in which they are drawn. Let Xj be the label of the jth draw, 1 ≤ j ≤ m. Assume that all orderings of the m draws are equally likely. Fix two distinct labels a, b ∈ {1, . . . , m}. Let

N = min{n ∈ {1, . . . , m} : Xn ∈ {a, b}}

be the index of the first draw that is either a or b. Let Y = XN ∈ {a, b} be the label of this draw. Find the joint PMF of (N, Y ) and the marginal PMFs of N and Y . Are N and Y independent random variables?

Hint: You need to find P (N = n, Y = y) for all possible values n, y first.

Answer 1