Question

Suppose that a die is rolled twice. What are the possible values that the following random variables can take on:

(a) the maximum value to appear in the two rolls;

(b) the minimum value to appear in the two rolls;

(c) the sum of the two rolls;

(d) the value of the first roll minus the value of the second roll?

Short answer

1. The maximum value to appear in two rolls are $\{1,2,3,4,5,6\}$
2. The minimum value to appear in the two rolls are $\{1,2,3,4,5,6\}$
3. The sum of the two rolls can be from $2to12$
4. The value of the first roll minus the value of the second roll from$-5to5$

Step-by-step solution

Step1: Given Information (part a)

Let's consider that a die is rolled twice.

The maximum of two rolls is also remains a random variable.

Step2: Explanation (part a)

A die includes the numbers of $\{1,2,3,4,5,6\}$. That means there are $36$combinations can occur.

Since for every $k\in \{1,2,3,4,5,6\}$ordered pair $(k,k)$can occur.

Therefore, the maximum value appear in the two rolls is$\{1,2,3,4,5,6\}$

Final Answer 

The maximum value appear in the two rolls is $\{1,2,3,4,5,6\}$

Step4: Given Information (part b)

Let's consider that a die is rolled twice.

The maximum of two rolls is also remains a random variable.

We have to find the minimum value to appear in the two rolls.

Step5: Explanation (part b)

Since,$(k,k)$can occur in two rolls, Where $k\in \{1,2,3,4,5,6\}$.

Therefore, the minimum value to appear in the two rolls are$\{1,2,3,4,5,6\}$

Final Answer (Part b)

The minimum value to appear in the two rolls; $\left\{1,2,3,4,5,6\right\}$

Step7: Given Information (part c)

Let's consider that a die is rolled twice.

The maximum of two rolls is also remains a random variable.

We have to find the sum of two rolls.

Step8: Explanation (part c)

Both die can take values from $1$to $6$. Therefore the smallest sum is $2$and the largest sum is $12$.It is trivial to see that every number un between can occur.

The possible values are$\left\{2,3,4,5,6,7,8,9,10,11,12\right\}$

Final Result (part c)

The sum of two rolls are taken from$2$to$12$

Step10: Given Information (part d)

Let's consider that a die is rolled twice.

The maximum of two rolls is also remains a random variable.

We have to find the value of the first roll minus the value of the second roll.

Step11: Explanation (part d)

Possible value for first roll $\left\{1,2,3,4,5,6\right\}$

Possible value for second roll$\left\{1,2,3,4,5,6\right\}$

The value of first roll minus second roll

$\left\{-5,-4,-3,-2,-1,0,1,2,3,4,5\right\}$

Step12: Final Result (part d)

The value of the first roll minus the value of the second roll from$-5to5$