FRM Part I · FRM Exam Part I · Random Variables
A continuous random variable X has density f(x) = 3x² for 0 ≤ x ≤ 1, and zero elsewhere. What is the variance of X?
The variance is 0.0375. The mean is 3/4 and the second moment is 3/5, so variance equals 0.60 minus 0.75 squared, or 0.60 - 0.5625 = 0.0375.
- A0.0375Correct
- B0.0600
- C0.0750
- D0.1875
Explanation
E[X] = ∫3x³ dx = 3/4 = 0.75. E[X²] = ∫3x⁴ dx = 3/5 = 0.60. Variance = 0.60 - 0.5625 = 0.0375. Option 0.0750 is double the correct figure and 0.1875 is 0.75 less 0.5625 misapplied, i.e. not a valid step; 0.0600 is E[X²]/10 with no basis.
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