Q.7.37 - A First Course in Probability

Question

A die is rolled twice. Let X equal the sum of the outcomes, and let Y equal the first outcome minus the second.

Compute $Cov(X, Y).$

Step-by-Step Solution

Answer

The value of $Cov(X, Y)$ is $0.$

Step 1: Given Information

$X=$The sum of the outcomes,

$Y=$The first outcome minus the second

$Cov(X,Y)=?$

Step 2: Explanation

Let $X$ denote equal the sum of the outcome.

Let $Y$ denote equal the first outcome minus the second.

Let $Z_1$be the outcome of the first role.

Let $Z_2$ be the outcome of the second toll.

Thus, $X=Z_1+Z_2$ and $Y=Z_1-Z_2$

Step 3: Explanation

Calculate the covariance of the $Cov(X, Y)$

$Cov(X,Y)=Cov\left(Z_1+Z_2,Z_1-Z_2\right)$

$=Cov\left(Z_1,Z_1\right)-Cov\left(Z_1,Z_2\right)+Cov\left(Z_2,Z_1\right)-Cov\left(Z_2,Z_2\right)$

$=Cov\left(Z_1,Z_1\right)-Cov\left(Z_2,Z_2\right)$

$=Var\left(Z_1\right)-Var\left(Z_2\right)$ Since, $Var\left(Z_1\right)=Var\left(Z_2\right)$

$=0$

Here, $Var\left(Z_1\right)=Var\left(Z_2\right)$because, $Z_1$ and $Z_2$are identically distributed.

Step 4: Final Answer

The value of $Cov(X,Y)$ is $0.$