Let X1 and X2 be the number on two independent fairdie rolls

Let X1 and X2 be the number on two independent fair-die rolls. Let X be the minimum and Y the maximum of X1 and X?. Find the conditional distribution of Y given X = x, x = 1,2,... ,6. Compute E(Y|X = x). Find the conditional distribution of X given Y y, y = 1,2,... ,6. Compute E(X|Y = y).

Solution

The joint distribution of X=min[X1,X2] and Y =max[X1,X2] then

  Conditional distribution of Y given X=x,x=1,2,3,4,5,6 is

  Expectation of Y given X=x,x=1,2,3,4,5,6 is

  

Conditional distribution of X given Y=y,y=1,2,3,4,5,6 is

  Expectation of X given Y=y,y=1,2,3,4,5,6 is

  

X\\Y	1	2	3	4	5	6	p(X)
1	1/36	1/18	1/18	1/18	1/18	1/18	11/36
2	0	1/36	1/18	1/18	1/18	1/18	9/36
3	0	0	1/36	1/18	1/18	1/18	7/36
4	0	0	0	1/36	1/18	1/18	5/36
5	0	0	0	0	1/36	1/18	3/36
6	0	0	0	0	0	1/36	1/36
p(Y)	1/36	3/36	5/36	7/36	9/36	11/36	1