Question

A fair die is rolled twice. Let X be the maximum of the two rolls. Find the distribution of X.

Let Y be the minimum of the two rolls. Find the variance of Y.

Answer 1

X can take values 1,2,3,4,5,6.

X=1: (1,1)

X=2: (1,2),(2,1),(2,2)

X=3: (1,3),(3,1),(2,3),(3,2),(3,3)

X=4: (1,4),(4,1),(2,4),(4,2),(3,4),(4,3),(4,4)

X=5: (1,5),(5,1),(2,5),(5,2),(3,5),(5,3),(4,5),(5,4),(5,5)

X=6: (1,6),(6,1),(2,6),(6,2),(3,6),(6,3),(4,6),(6,4),(5,6),(6,5),(6,6)

Total number of observations = 62 = 36

So, probability distribution of X:

X	1	2	3	4	5	6
P(X)

Y can take values 1,2,3,4,5,6.

Y=1: (1,1),(1,2),(2,1),(1,3),(3,1),(1,4),(4,1),(1,5),(5,1),(1,6),(6,1)

Y=2: (2,2),(2,3),(3,2),(2,4),(4,2),(2,5),(5,2),(2,6),(6,2)

Y=3: (3,3),(3,4),(4,3),(3,5),(5,3),(3,6),(6,3)

Y=4: (4,4),(4,5),(5,4),(4,6),(6,4)

Y=5: (5,5),(5,6),(6,5)

Y=6: (6,6)

Total number of observations = 62 = 36

So, probability distribution of Y:

Y	1	2	3	4	5	6
P(Y)

Now,

So,