Probability and Voting: Covariance of Republican and Democratic Votes

[Kalinka35]

Homework Statement

In a town there are r+1 Republicans and d+1 Democrats. There is a Republican candidate and a Democratic candidate, both of whom will vote for themselves. Aside from them, a Republican voter will vote for a Democrat with probability p_RD and a Democrat will vote for a Republican with probability p_DR. Assume the voters behave independently.
What is the covariance of the number of votes the Republican receives and the number of votes the Democrat receives?

Homework Equations

Cov(X,Y) = E(XY) - E(X)E(Y)

The Attempt at a Solution

I let X=votes for Republican, Y=votes for Democrat.
I got that E(X)=1+d*p_DR + r(1-p_RD) and
E(Y) = 1+r*p_RD + d(1-p_DR)
But how do you get E(XY). I was thinking of representing X as a sum of Bernoulli's (1 if the vote is for the Republican, 0 if for the Democrat), but I still don't see how I would get E(XY) with that approach.

Thanks.

 

[mighty2000]

To calculate E(XY), you can use the fact that the votes for the Republican and Democrat are independent and that a Bernoulli random variable has a variance equal to its probability of success. Therefore, you can write:

E(XY) = E(X)E(Y) + Cov(X,Y)

Since the votes are independent, E(X)E(Y) = (1+r*pRD + d(1-pDR))(1+d*pDR + r(1-pRD)) = 1 + r*pRD + d*pDR + r*d*pRD*(1-pRD) + r*d*pDR*(1-pDR)

Next, you can use the definition of covariance to write:

Cov(X,Y) = E(XY) - E(X)E(Y) = 1 + r*pRD + d*pDR + r*d*pRD*(1-pRD) + r*d*pDR*(1-pDR) - (1+r*pRD + d(1-pDR))(1+d*pDR + r(1-pRD))

Simplifying this expression will give you the covariance of the number of votes the Republican receives and the number of votes the Democrat receives.

I hope this helps. Let me know if you have any further questions.