Variance of a sum of three equally correlated quantities

Question

Three quantities X, Y, and Z each have variance 4, and every pair has covariance 1: that is, \mathrm{Cov}(X,Y) = \mathrm{Cov}(X,Z) = \mathrm{Cov}(Y,Z) = 1. You form the total T = X + Y + Z. What is the variance of the total?

Accepted Answer

For a sum of several quantities, the variance is the sum of all the individual variances plus twice the covariance of every distinct pair: \mathrm{Var}(T) = \underbrace{\mathrm{Var}(X) + \mathrm{Var}(Y) + \mathrm{Var}(Z)}{	ext{three variances}} + 2\big[\underbrace{\mathrm{Cov}(X,Y) + \mathrm{Cov}(X,Z) + \mathrm{Cov}(Y,Z)}{	ext{three pairs}}\big]. In words: add up each variance, then add twice each pairwise covariance. The main thing to get right is the count of pairs: three quantities give three distinct pairs. Apply it. \mathrm{Var}(T) = (4 + 4 + 4) + 2(1 + 1 + 1) = 12 + 6 = 18. The standa…

Variance of a sum of three equally correlated quantities

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