Suppose we roll two fair sixsided dice Let X be the smaller

Suppose we roll two fair six-sided dice. Let X be the smaller of the two numbers appearing, and Y be the larger (if the two numbers are the same, so are X and Y). Find the joint pmf of X and Y.

Solution

The joint mass function of X and Y can be:

P(1,1)=p[x=1,y=1]=1/36

P(1,2)= p[x=1,y=2]=2/36

P(1,3)= p[x=1,y=3]=2/36

P(1,4)= p[x=1,y=4]=2/36

P(1,5)= p[x=1,y=5]=2/36

P(1,6)= p[x=1,y=6]=2/36

P(2,2)= p[x=2,y=2]=1/36

P(2,3)= p[x=2,y=3]=2/36

P(2,4)= p[x=2,y=4]=2/36

P(2,5)= p[x=2,y=5]=2/36

P(2,6)= p[x=2,y=6]=2/36

P(3,3)= p[x=3,y=3]=1/36

P(3,4)= p[x=3,y=4]=2/36

P(3,5)= p[x=3,y=5]=2/36

P(3,6)= p[x=3,y=6]=2/36

P(4,4)= p[x=4,y=4]=1/36

P(4,5)= p[x=4,y=5]=2/36

P(4,6)= p[x=4,y=6]=2/36

P(5,5)= p[x=5,y=5]=1/36

P(5,6)= p[x=5,y=6]=2/36

P(6,6)= p[x=6,y=6]=1/36