Two fair dice are rolled Let X denote the largest of the two

Two fair dice are rolled. Let X denote the largest of the two values and let Y
denote the smallest.
(a) Find the joint and marginal probability mass functions of X and Y .
(b) Find the conditional probability mass function of Y given x = 3.
(c) Find the conditional probability that X > 3 given that Y < 3.
(d) Find the probability mass function of Z = X - Y .
(e) Are X and Y independent?

Solution

This can be done by direct calculation. The joint pmf is given by the values of the following table divided by 36:

y\\x 1 2 3 4 5 6 fY (y)

1 1 2 2 2 2 2 11

2 0 1 2 2 2 2 9

3 0 0 1 2 2 2 7

4 0 0 0 1 2 2 5

5 0 0 0 0 1 2 3

6 0 0 0 0 0 1 1

fX(x) 1 3 5 7 9 11 36

Summing over each column, we get that the pmf of X is given by the last row divided by 36. Summing over each row, we get that the pmf of Y is given by the last column divided by 36.

(b) The conditional pmf of Y given x = 3 is

fY |X(y|x = 3) = P(Y = y|X = 3) = P(X = 3, Y = y)/ P(X = 3)

This is equal to the column corresponding to x = 3 divided by the total at the bottom of the column, and is given by the following table:

y 1 2 3 4 5 6

fY |X(y|x = 4) 2/5 2/5 1/5 0 0 0

c) Summing the appropriate terms over the table, the given probability is equal to P(X > 3|Y < 3) =

P(X > 3, Y < 3)/P(Y < 3) = 12/ 20 = 3/ 5

d) Z is constant along the diagonals of the table of part (a). Summing over each diagonal, we get that the pmf of Z is given by the values of the following table divided by 36:

z 0 1 2 3 4 5

6 10 8 6 4 2

which is equal to the following table divided by 18:

z 0 1 2 3 4 5

3 5 4 3 2 1

e)no they are not independent