Covariance and correlation with given probabilities

Question

In a football tournament you play each of the other ten teams once. In every match you get 3 points if you win, 1 point for a draw and 0 points if you lose. For each match the probability you win is 0.6, the probability you draw is 0.1 and the probability you lose is 0.3, independently of the results of all other matches.

Let W be the number of games you win and L be the number of games you lose. What is the covariance and correlation between W and L

1. i am stuck with this problem, any help would be appreciated.

Answer 1

cov(W,L) = p(10)(10-6)(0-3) + p(9)(9-6)(1-3)...p(0)(0-6)(10-3)

corr(W,L) = cov(W,L)/sigma(W)sigma(L)

sigma^2(W) = p(10)(10-6)^2 + p(9)(9-6)^2 + ... + p(0)(0-6)^2

similarly for sigma^2(L)

where p(n) is the binomial probability for n of 10

This is a lot of computation, but not too bad if you use a spreadsheet