FRM Part I · FRM Exam Part I · Common Univariate Random Variables
A return is drawn from component A with probability 0.75 (mean 1, standard deviation 2) and from component B with probability 0.25 (mean -2, standard deviation 4), in percent. What is the variance of the mixture?
The mixture variance is 8.6875. Compute the mixture mean of 0.25 and the second moment of 8.75 from each component's variance plus squared mean, then subtract the squared mean. Using only within-component variance gives 7.0, which omits the effect of different means.
- A8.6875Correct
- B8.75
- C7.0
- D1.6875
Explanation
The mixture mean is 0.75(1) + 0.25(-2) = 0.25. The second moment is 0.75(4+1) + 0.25(16+4) = 3.75 + 5 = 8.75. Variance = 8.75 - 0.25^2 = 8.6875. The value 8.75 forgets to subtract the squared mean. The value 7.0 is only the within-component variance, and 1.6875 is only the between-component part.
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