Why yes-votes and no-votes are perfectly negatively correlated

Question

A committee of 20 people each vote yes or no. Let X be the number of yes votes and Y the number of no votes, so X + Y = 20 always. Suppose X has variance 5. What is the correlation between X and Y?

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The key is that the two counts must add up to a fixed 20, so Y = 20 - X. Whenever two quantities are tied together by an exact straight-line rule like this, their correlation is either +1 or -1, and the minus sign here comes from the fact that Y goes down by one for every extra yes. Covariance. Write Y = 20 - X and expand: \mathrm{Cov}(X, Y) = \mathrm{Cov}(X,\, 20 - X) = \mathrm{Cov}(X, 20) - \mathrm{Cov}(X, X) = 0 - \mathrm{Var}(X) = -5. The constant 20 contributes nothing, and the covariance of X with itself is its variance. Standard deviations. Since Y = 20 - X is just X flipped and shifte…

Why yes-votes and no-votes are perfectly negatively correlated

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