Question

Roll two dice at the same time, and define a random variable X as the sum of the two faces observed. Determine the CDF and PMF of X. Sketch the CDF.

Answer 1

Solution:

Given: Two dice are rolled at the same time.

X = the sum of the two faces observed

We have to determine the CDF and PMF of X and sketch the CDF.

Sample Space for rolling of two dice is:

Sum of faces means we add two numbers observed on faces.

Suppose if we get (1,1) , then sum = 1+1 = 2

if we get (3,5) , then Sum = 3+5

So minimum sum is 1+1 = 2 and maximum sum is 6+6 = 12

thus we will have sum of faces from 2 to 12

Thus we need to make following table:

To get PMF , P(X = x) , we divide each frequency by N = 36

That is: P( X=2) = 1/36 = 0.0278

P(X=3) = 2/36=0.0556 and so on.

Outcomes	X = Sum of faces	f : Frequency	PMF P(X=x)	Calculations of CDF	CDF      P(X ≤ x )
(1,1)	2	1	0.0278	=0.0278 =	0.0278
(1,2) , ( 2 , 1)	3	2	0.0556	=0.0278+0.0556 =	0.0834
(1,3) , (2,2) , ( 3 ,1)	4	3	0.0833	=0.0834+0.0833=	0.1667
(1,4) , (2,3) , ( 3 ,2) , (4,1)	5	4	0.1111	=0.1667+0.1111=	0.2778
(1,5) , (2,4) , ( 3 ,3) , (4,2) ,(5,1)	6	5	0.1389	=0.2778+0.1389=	0.4167
(1,6) , (2,5) , ( 3 ,4) , (4,3) ,(5,2),(6,1)	7	6	0.1667	=0.4167+0.1667=	0.5834
(2,6) , (3,5) , ( 4,4) , (5,3) ,(6,2)	8	5	0.1389	=0.5834+0.1389=	0.7222
(3,6), (4,5) , (5,4) , (6,3)	9	4	0.1111	=0.7222+0.1111=	0.8334
(4,6) , ( 5,5) , ( 6,4)	10	3	0.0833	=0.8334+0.0833=	0.9167
(5,6) , (6,5)	11	2	0.0556	=0.9167+0.0556=	0.9722
(6,6)	12	1	0.0278	=0.9722+0.0278=	1.0000
		N = 36

To sketch CDF, take x values on X axis and CDF values on Y axis