FRM Part I · FRM Exam Part I · Random Variables
A continuous random variable has density f(x) = 0.5x for 0 ≤ x ≤ 2 and 0 elsewhere. What is the variance of X?
The mean is 4/3 and the second moment is 2. Variance equals E[X squared] minus the squared mean, so 2 - 16/9 = 2/9. The value 4/3 is only the mean and not the variance.
- A2/9Correct
- B4/3
- C8/9
- D1/2
Explanation
E[X] = integral of x(0.5x) from 0 to 2 = 0.5(8/3)/... compute: 0.5 times x^3/3 at 2 = 0.5(8/3) = 4/3. E[X^2] = integral of 0.5x^3 = 0.5(16/4) = 2. Variance = 2 - (4/3)^2 = 2 - 16/9 = 2/9. Option 4/3 is the mean, and 8/9 comes from forgetting to square properly in the subtraction.
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