SOLUTION: Consider the random experiment of tossing two fair dice and recording the up faces. Let X be the sum of the two dice, and let Y be the absolute value of the difference of the two d

Question 1199109: Consider the random experiment of tossing two fair dice and recording the up faces. Let X be the sum of the two dice, and let Y be the absolute value of the difference of the two dice. 1.what is the probability function of Y?
2.What is the cumulative distribution function of Y?

Answer by Edwin McCravy(20056) (Show Source): You can put this solution on YOUR website!

There are 36 way a pair of dice can fall:

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

for x=0, the rolls with absolute difference 0 are these 6: 
(1,1), (2,2), (3,3), (4,4), (5,5), (6,6).  so P(0) = 6/36 = 1/6. 

for x=1, the rolls with absolute difference 1 are these 10: 
(1,2), (2,3), (3,4), (4,5), (5,6), (2,1), (3,2), (4,3), (5,3), (6,5).  
So P(1) = 10/36 = 5/18.

for x=2, the rolls with absolute difference 2 are these 8: 
(1,3), (2,4), (3,5), (4,6), (6,4), (5,3), (4,2), (3,1).  
So P(2) = 10/36 = 5/18.

for x=3, the rolls with absolute difference 3 are these 6: 
(1,4), (2,5), (3,6), (4,1), (5,2), (4,1).  
So P(3) = 6/36 = 1/6.

for x=4, the rolls with absolute difference 4 are these 4: 
(1,5), (2,6), (5,1), (6,2).  
So P(4) = 4/36 = 1/9.

for x=5, the rolls with absolute difference 5 are these 2: 
(1,6), (1,6).  
So P(5) = 2/36 = 1/18.

So the probability function of Y is 

  x   P(x) 
  0   6/36 = 1/6 
  1  10/36 = 5/18
  2   8/36 = 2/9
  3   6/36 = 1/6
  4   4/36 = 1/9
  5   2/36 = 1/18
-----------------
     36/36 = 1

The cumulative distribution function is found by accumulating
value plus the sum of the values before it.
  x   CP(x) 
  0   6/36 = 1/6 
  1   6/36+10/36 = 16/36 = 4/9 
  2   8/36+6/36+8/36 = 22/36 = 11/18
  3   6/36+8/36+6/36+6/36 = 26/39 = 2/3
  4   6/36+8/36+6/36+10/36+4/36 = 34/36 = 17/18
  5   6/36+8/36+6/36+10/36+4/36+2/36 = 36/36 = 1
-----------------
     36/36 = 1

Edwin

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