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FRM Part I · FRM Exam Part I · Fundamentals of Probability

Random variables X and Y have Var(X) = 25, Var(Y) = 16 and Cov(X,Y) = 10. What is the correlation between (X + Y) and (X - Y), to two decimals?

The correlation is about 0.25. The covariance of the sum and difference equals Var(X) minus Var(Y), which is 9. The variances of the sum and difference are 61 and 21, so the correlation is 9 divided by the square root of 1281, roughly 0.25.

  1. A0.25Correct
  2. B0.43
  3. C0.15
  4. D0.36

Explanation

Var(X+Y) = 25+16+20 = 61. Var(X-Y) = 25+16-20 = 21. Cov(X+Y, X-Y) = Var(X) - Var(Y) = 9. Correlation = 9/√(61×21) = 9/√1281 ≈ 9/35.79 ≈ 0.25. The 0.43 figure divides by only 21 and 0.15 by only 61, while 0.36 divides by 25; none uses both standard deviations.

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