FRM Part I · FRM Exam Part I · Common Univariate Random Variables
A return distribution is a mixture: with probability 0.5 it is normal with mean 2% and standard deviation 1%, and with probability 0.5 it is normal with mean -2% and standard deviation 1%. What is the variance of the mixture (in %²)?
The mixture variance is 5 (%²). It equals the average within-component variance of 1 plus the variance of the component means around the overall mean of zero, which is 4, because the means are +2% and -2% with equal weights.
- A1
- B4
- C5Correct
- D3
Explanation
Variance = weighted average of component variances plus variance of the means. Average variance = 1. Mean of mixture = 0; variance of component means = 0.5 x 4 + 0.5 x 4 = 4. Total = 1 + 4 = 5. Choosing 1 ignores the dispersion of the means.
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