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

jernplate8

jernplate8

Answered question

2021-08-15

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

Answer & Explanation

irwchh

irwchh

Skilled2021-08-16Added 102 answers

Define Xito be indicator random variable that is equal to one if and only if we have obtained 1 in the ith throw.
Similary, define Yi to be indicator random variable that is equal to one if and only if we have obtained 2 in ith throw. Therefore
X=i=1nXi,Y=j=1nYj
Using basic properties of the covariance, we have that
Cov(X,Y)=Cov(X=i=1nXi,Y=j=1nYj)=ijCov(Xi,Yj)
Observe that for ij random variablesXi and Yj are independent.
Knowing what happened in ith throw does not change probabilities for jth throw.
Hence, in that case
Cov(Xi,Yj)=0
So
Cov(X,Y)=iCov(Xi,Yi)
We have that
Cov(Xi,Yi)=E(Xi,Yi)E(Xi)E(Yi)
Observe that P(XiYi=1)=P(Xi=1,Yi=1)=0since it is impossible have two outcomes in a single throw simultaneously. Hence
Cov(Xi,Yi)=136
which implies
Cov(X,Y)=n36
Result: Cov(X,Y)=n36

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