Two fair dice are rolled. Find the joint probability mass function of X and Y when 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.
Two fair dice are rolled. Find the joint probability mass function of X and Y when 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.
Two fair dice are rolled. Find the joint probability mass function of X and Y when
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.
Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.
(a)
Expert Solution
To determine
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:
P(X=k,Y=l)={236k<l<2k136l=2k
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
X=max(N1,N2)
And
Y=N1+N2
Thus,
X∈{1,...,6}
And
Y∈{2,...,12}
Also,
We have
X<Y almost certainly.
Then
Take any k<l ,
Where,
k and l are from the ranges of X and Y .
Consider event
X=k,Y=l .
That means
The maximum value on any die is k .
And
The sum of both dice is l .
Now,
Observe that
If l<2k ,
The only possible pairs of (N1,N2) corresponding to the event are (l,l−k) and (l−k,k) .
If l=2k ,
The only possible pair is (k,k) .
Therefore,
The required probability mass function is
P(X=k,Y=l)={236k<l<2k136l=2k
(b)
Expert Solution
To determine
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:
P(Y=l|X=k)P(X=k)={k36k=l136k<l
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
X=N1
And
Y=max(N1,N2)
Then
Observe that
{1,...,6}
And
X≤Y almost certainly.
Take any k≤l from the range {1,...,6}
Then
We have
P(X=k,Y=l)=P(Y=l|X=k)P(X=k)
Suppose that
k=l
We already have
X=k
In such case,
N2 can be any number from the range 1,...,k to obtain the required Y=l .
Thus,
P(Y=l|X=k)P(X=k)=k6⋅16=k36
If k<l ,
And
X=k ,
N2 must be equal to l to obtain Y=l .
Thus,
P(Y=l|X=k)P(X=k)=16⋅16=136
(c)
Expert Solution
To determine
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:
P(X=k,Y=l)={236k<l136k=l
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
X=min(N1,N2)
And
Y=max(N1,N2)
We also have
X≤Y almost certainly.
Then
Take any k≤l .
Suppose that
k<l
In such case,
We need to have
N1=k,N2=l
Or
N1=l,N2=k
Thus,
There are only two possibilities.
P(X=k,Y=l)=236
If k=l ,
The only possibility will be (N1,N2)=(k,k) .
Thus,
P(X=k,Y=l)=136
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