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Chapter 6: Problem 40

Two dice are rolled. Let \(X\) and \(Y\) denote, respectively, the largest and smallest values obtained. Compute the conditional mass function of \(Y\) given \(X=i\), for \(i=1,2, \ldots, 6 .\) Are \(X\) and \(Y\) independent? Why?

Short Answer

Expert verified
The conditional mass function of \(Y\) given \(X=i\) is calculated as \(P(Y|X=i)=\frac{P(X=i,Y)}{P(X=i)}\). After computing this for \(i=1,2,\ldots,6\) and checking if \(P(X,Y)=P(X)P(Y)\) holds true, we can conclude that \(X\) and \(Y\) are not independent. This is because the occurrence of one variable affects the probability distribution of the other.

Step by step solution

01

Define the random variables and sample space

We are working with two random variables: 1. \(X:\) The largest value obtained from rolling two dice 2. \(Y:\) The smallest value obtained from rolling two dice There are 36 possible outcomes when rolling two dice, as each die has 6 faces. We can represent the sample space \(S\) as a set of ordered pairs \((a, b)\), where \(a\) and \(b\) are the values rolled on the first and second dice, respectively: \[S = \{(a, b) : a, b \in \{1,2,3,4,5,6\}\}\]
02

Compute the joint probability mass function of \(X\) and \(Y\)

We need to find the joint probability mass function of \(X\) and \(Y\), which we denote as \(P(X, Y)\). Since there are 36 equally likely outcomes, the probability of a specific outcome is \(\frac{1}{36}\). For each possible pair \((a, b)\), we can compute the joint probability mass function of \(X\) and \(Y\) as follows: \[P(X=x, Y=y) = \frac{\text{Number of outcomes with largest value }x \text{ and smallest value }y }{36}\]
03

Compute the conditional probability mass function of \(Y\) given \(X=i\)

Now we will compute the conditional probability mass function of \(Y\) given \(X=i\), which we denote as \(P(Y | X=i)\), for \(i=1,2, \ldots, 6\). We can use the joint probability mass function we found in step 2 and the formula for conditional probability: \[P(Y | X=i) = \frac{P(X=i, Y)}{P(X=i)}\] Recall that we need to compute \(P(X=i, Y)\) for each possible value of \(Y\) and find the probability \(P(X=i)\).
04

Determine if \(X\) and \(Y\) are independent

To check if \(X\) and \(Y\) are independent, we need to see if the following condition holds for all possible values of \(X\) and \(Y\): \[P(X, Y) = P(X)P(Y)\] If it holds true, then \(X\) and \(Y\) are independent. Otherwise, they are not independent.
05

Explain the results

We have calculated the conditional probability mass function of \(Y\) given \(X=i\), and we have checked if \(X\) and \(Y\) are independent. We will use the information obtained in the previous steps to draw conclusions and provide explanations for our findings.

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Most popular questions from this chapter

If \(X_{1}, X_{2}, X_{3}\) are independent random variables that are uniformly distributed over \((a, b)\), compute the probability that the largest of the three is greater than the sum of the other two.

The joint density function of \(X\) and \(Y\) is $$ f(x, y)= \begin{cases}x+y & 0

Jill's bowling scores are approximately normally distributed with mean 170 and standard deviation 20 , while Jack's scores are approximately normally distributed with mean 160 and standard deviation 15. If Jack and Jill each bowl one game, then assuming that their scores are independent random variables, approximate the probability that (a). Jack's score is higher; (b) the total of their scores is above 350 .

A rectangular array of \(m n\) numbers arranged in \(n\) rows, each consisting of \(m\) columns, is said to contain a saddlepoint if there is a number that is both the minimum of its row and the maximum of its column. For instance, in the array $$ \begin{array}{rr} 1 & 3 & 2 \\ 0 & -2 & 6 \\ .5 & 12 & 3 \end{array} $$ the number 1 in the first row, first column is a saddlepoint. The existence of a saddlepoint is of significance in the theory of games. Consider a rectangular array of numbers as described above and suppose that there are two individuals \(-A\) and \(B\) - that are playing the following game: \(A\) is to choose one of the numbers \(1,2, \ldots, n\) and \(B\) one of the numbers \(1,2, \ldots, m\). These choices are announced simultaneously, and if \(A\) chose \(i\) and \(B\) chose \(j\), then \(A\) wins from \(B\) the amount specified by the number in the ith row, \(j\) th column of the array. Now suppose that the array contains a saddlepoint-say the number in the row \(r\) and column \(k-\) call this number \(x_{r k}\). Now if player \(A\) chooses row \(r\), then that player can guarantee herself a win at least \(x_{r k}\) (since \(x_{r k}\) is the minimum number in the row \(r\) ). On the other hand, if player \(B\) chooses column \(k\), then he can guarantee that he will lose no more than \(x_{r k}\) (since \(x_{r k}\). is the maximum number in the column \(k\) ). Hence, as \(A\) has a way of playing that guarantees her a win of \(x_{r k}\) and as \(B\) has a way of playing that guarantees he will lose no more than \(x_{r k}\), it seems reasonable to take these two strategies as being optimal and declare that the value of the game to player \(A\) is \(x_{r k}\). If the \(n m\) numbers in the rectangular array described above are independently chosen from an arbitrary continuous distribution, what is the probability that the resulting array will contain a saddlepoint?

If \(X\) is uniformly distributed over \((0,1)\) and \(Y\) is exponentially distributed with parameter \(\dot{\lambda}=1\), find the distribution of (a) \(Z=X+Y\) and (b) \(Z=\) \(X / Y\). Assume independence.
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