Question

Consider some 8-sided dice. Roll two of these dice. Let X be the minimum of the two values that appear. Let Y denote the maximum.  

a) Find the joint mass p_X,Y (x,y).

b) Compute p_X│Y (x│y) in all cases. Express your final answer in terms of a table.

Answer 1

Answer :

a)Given that :

Min of two values that appear = X

Maximum = Y

given dices through = 8

then we get the no.of possibilities = 8 * 8 = 64

there are :(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (2,7) (2,8)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (3,7) (3,8)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (4,7) (4,8).................and

..........................................(8,8)

therefore,now p(X,Y) = 0 where X>Y

and p(X,Y) = 1/64 where X = Y

p(X,Y) = 2/64 = 1/32 where X<Y

for X = 1,2,3,4,5,6,7,8

and Y = 1,2,3,4,5,6,7,8

b)Now consider the joint mass probability function of p(X/Y)

p(X/Y)	1	2	3	4	5	6	7	8	Sum
1	0.0156	0.0312	0.0312	0.0312	0.0312	0.0312	0.0312	0.0312	0.234
2		0.0156	0.0312	0.0312	0.0312	0.0312	0.0312	0.0312	0.203
3			0.0156	0.0312	0.0312	0.0312	0.0312	0.0312	0.172
4				0.0156	0.0312	0.0312	0.0312	0.0312	0.140
5					0.0156	0.0312	0.0312	0.0312	0.109
6						0.0156	0.0312	0.0312	0.078
7							0.0156	0.0312	0.047
8								0.0156	0.0156
Sum	0.156	0.0468	0.078	0.109	0.140	0.172	0.203	0.234	1