Q.7.36 - A First Course in Probability

Question

Let $X$ be the number of $1’s$ and $Y$ the number of $2\text{'}s$ that occur in $n$ rolls of a fair die. Compute $Cov(X, Y)$.

Step-by-Step Solution

Answer

The value of $Cov(X,Y)$ is $\frac{-n}{36}.$

Step 1: Given Information

Number of $1\text{'}s=X$

Number of $2\text{'}s=Y$ 

Fair die rolls$=n$

Step 2: Explanation

Calculate the covariance of$(X,Y)$

$X=$Number of 1's in $n$ rolls of a fair Die

$Y=$Number of 2's in $n$ rolls of A fair Die

$\therefore \mathrm{E}\left(\mathrm{X}\right)=\frac{n}{6}$

$\mathrm{E}\left(\mathrm{Y}\right)=\frac{n}{6}$

$\mathrm{E}\left(\mathrm{XY}\right)=\frac{n}{6}\times \frac{(n-1)}{6}$

$=\frac{n(n-1)}{36}$

$\therefore Cov(X,Y)=E\left(XY\right)-E\left(X\right)·E\left(Y\right)$

Step 3: Explanation

Compute$Cov(X,Y)$

$=\frac{n}{6}·\frac{n-1}{6}-\frac{n}{6}·\frac{n}{6}$

$=\left(\frac{n^2}{36}-\frac{n}{36}\right)-\frac{n^2}{36}$

$=\frac{-n}{36}$

Step 4: Final Answer

Hence, the value of $Cov(X,Y)$ is $\frac{-n}{36}.$