
Concept explainers
Two fair dice are rolled. Find the joint
a. X is the largest value obtained on any die and
Y is the sum of the values;
b. X is the value on the first die and
Y is the larger of the two values;
c. X is the smallest and
Y is the largest value obtained on the dice.
(a)

To find: Joint probability mass function with X be the largest value Y be the sum of values.
Answer to Problem 6.1P
Joint probability mass function:
Explanation of Solution
Given information:
While rolling two fair dice,
X is the largest value obtained on any die.
Y is the sum of the values.
Let
N1 and N2 as the random variables that mark numbers obtained on the first and second die.
We know that
N1 and N2 are independent.
Such that
N1 , N2 ? D Unif(1,…,6).
In this part,
We have
And
Thus,
And
Also,
We have
Then
Take any
Where,
k and l are from the ranges of X and Y .
Consider event
That means
The maximum value on any die is k .
And
The sum of both dice is l .
Now,
Observe that
If
The only possible pairs of
If
The only possible pair is
Therefore,
The required probability mass function is
(b)

To find: Joint probability mass function with X be the value on first die and Y be the larger value.
Answer to Problem 6.1P
Joint probability mass function:
Explanation of Solution
Given information:
While rolling two fair dice,
X is the value on the first die.
Y is the larger of the two values.
Let
N1 and N2 as the random variables that mark numbers obtained on the first and second die.
We know that
N1 and N2 are independent.
In this part,
We have
And
Then
Observe that
And
Take any
Then
We have
Suppose that
We already have
In such case,
N2 can be any number from the range
Thus,
If
And
N2 must be equal to l to obtain
Thus,
(c)

To find: Joint probability mass function with X be the smallest and Y be the largest value obtained.
Answer to Problem 6.1P
Joint probability mass function:
Explanation of Solution
Given information:
While rolling two fair dice,
X is the smallest value.
Y is the largest value.
Let
N1 and N2 as the random variables that mark numbers obtained on the first and second die.
We know that
N1 and N2 are independent.
In this part,
We have
And
We also have
Then
Take any
Suppose that
In such case,
We need to have
Or
Thus,
There are only two possibilities.
If
The only possibility will be
Thus,
Want to see more full solutions like this?
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