Question

A Ph.D. candidate in Statistics must form a dissertation committee consisting of four faculty mem- bers in the Department of Statistics. The department has 5 Assistant Professors, 7 Associate Pro- fessors and 6 Full Professors. Let Y1 and Y2 denote the number of Associate Professors and Full Professors, respectively, who were selected to serve on the committee.

(a) Find the joint probability mass function of Y1 and Y2. Provide your solution in table form and

as a function.

(b) Find the marginal probability mass functions of Y1 and Y2.

(c) Are Y1 and Y2 independent? Justify your answer.

(d) Find the conditional probability mass function of Y2|Y1 = 2

(e) Let F denote the joint cumulative distribution function of Y1 and Y2 . Calculate F (1, 2), F(0,0), F(−1,1) and F(5,6)

(f) FindP(1≤Y1 <3,1≤Y2 <4)

(g) Find P(Y2 ≤ 2|Y1 = 1).

Answer 1