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FRM Part I · FRM Exam Part I · Multivariate Random Variables

X and Y are random variables with correlation 0.6. Y is replaced by W = -3Y + 5, and X by V = 2X. Which statement about the correlation between V and W, and the effect of the changes, is correct?

The correlation becomes -0.6. Adding constants and multiplying by positive constants do not change correlation, while multiplying one variable by a negative number reverses its sign. Covariance, by contrast, is scaled by the product of the multipliers, here -6, and correlation can never exceed 1 in magnitude.

  1. ACorrelation is -0.6, because scaling by positive constants leaves correlation unchanged while a negative multiplier flips its signCorrect
  2. BCorrelation is -1.8, because the scale factors multiply the correlation
  3. CCorrelation is 0.6, because correlation is unaffected by any linear transformation
  4. DCorrelation is -0.6 and the covariance is unchanged, because the constant 5 offsets the scaling

Explanation

Corr(aX+b, cY+d) = sign(ac)·Corr(X,Y). Here a=2 and c=-3 give a negative sign, so the correlation is -0.6. Covariance scales by ac = -6, so it is not unchanged; correlation is bounded by 1 so -1.8 is impossible.

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