FRM Part I · FRM Exam Part I · Common Univariate Random Variables
Returns are drawn from a calm regime with probability 0.7 (mean 0, standard deviation 1) and from a stressed regime with probability 0.3 (mean 0, standard deviation 3). What is the variance of the mixture?
The mixture variance is 3.4. Since both regimes have mean zero, the variance equals the weighted average of the component variances, 0.7 times 1 plus 0.3 times 9. Averaging standard deviations or using equal weights gives incorrect results.
- A3.4Correct
- B1.6
- C2.56
- D5.0
Explanation
With equal means, mixture variance is the probability-weighted average of the component variances: 0.7(1) + 0.3(9) = 0.7 + 2.7 = 3.4. The value 1.6 averages the standard deviations, and squaring it gives 2.56, which is also wrong. The value 5.0 ignores the probabilities and uses a simple average of the variances.
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