Exercise 1.44. Two fair dice are rolled. Let X be the maximum of the two numbers and Y the minimum of the two numbers on the dice. (a) Find the possible values of X and the possible values of Y . (b) Find the probabilities P(X ≤ k) for all integers k. Find the probability mass function of X. Hint. Noticing that P(X = k) = P(X ≤ k) − P(X ≤ k − 1) can save you some work. (c) Find the probability mass function of Y .

a. The possible values of X are 2,3,4,5,6

   The possible values of Y are 1,2,3,4,5

b.  X  123456Total

 n            0      24681030

  p(x)0     1/15  2/153/15  4/15  1/3  1  

c.   X123456Total

     n108642030

  p(x)       1/3  4/15   3/152/15  1/15  0  1  

Step-by-step explanation:

Rolling two fair dice, we obtain the following tables :

 123456

11,11,21,31,41,51,6

22,12,22,32,42,52,6

33,13,23,33,43,53,6

44,14,24,34,44,54,6

55,15,25,35,45,55,6

66,16,26,36,46,56,6

Let X be the maximum of the two numbers on the dice, then

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

Let Y be the minimum of the two numbers on the dice, then

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

Finding the probability mass functions of X and Y which contains the possible values of X and Y and their associated probabilities, we have

X123456Total

n024681030

p(x)01/15   2/153/15  4/15  1/3  1  

and

X123456Total

n108642030

p(x)  1/34/15  3/152/15  1/15  0  1  

44

step-by-step explanation: