Question

Two dice are rolled. Let X be the random variable representing the result of the first die, and Y be the random variable representing the largest value rolled on either die. Describe the joint probability density function for X and Y .

Answer 1

Let X and Y are the two random variables.

Where X represents outcome of the first die

Y represents maximum of the both outcome

So values the random variable can take are as,

X = {1,2,3,4,5,6}

Y = {1,2,3,4,5,6}

Sample space for the given event is,

S = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),

(2,2), (2,3), (2,4), (2,5), (2,6),

(3,3), (3,4), (3,5), (3,6),

(4,4),(4,5),(4,6),

(5,5),(5,6),

(6,6) }

Follwoing is the distribution of the given radnom variables and event -

		Y = maximum of both
		1	2	3	4	5	6	Total
X	1	1/21	1/21	1/21	1/21	1/21	1/21	1/21
	2	0	1/21	1/21	1/21	1/21	1/21	2/21
	3	0	0	1/21	1/21	1/21	1/21	3/21
	4	0	0	0	1/21	1/21	1/21	4/21
	5	0	0	0	0	1/21	1/21	5/21
	6	0	0	0	0	0	1/21	6/21
	Total	1/21	2/21	3/21	4/21	5/21	6/21	1
	Sample space size = 21