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 $Cov(X,Y)$.

Short answer

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

Step-by-step solution

Given Information

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

Number of $2\text{'}s=Y$

Fair die rolls$=n$

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

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

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

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

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

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

Explanation

Compute$Cov(X,Y)$

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

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

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

Final Answer 

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