FRM Part I · FRM Exam Part I · Random Variables
A risk analyst models a loss ratio X with density f(x) = c(1 - x) for 0 ≤ x ≤ 1 and 0 otherwise. What is the value of c, and what is E[X]?
Normalising the density requires c times 1/2 to equal 1, so c = 2. The mean is 2 times (1/2 - 1/3) = 1/3. The density declines in x, so the mean sits below the midpoint of 0.5.
- Ac = 1; E[X] = 0.50
- Bc = 2; E[X] = 0.50
- Cc = 2; E[X] = 1/3Correct
- Dc = 1/2; E[X] = 1/3
Explanation
The density must integrate to 1: c times the integral of (1 - x) over 0 to 1 equals c(1/2), so c = 2. Then E[X] = 2 times the integral of x(1 - x), which is 2(1/2 - 1/3) = 1/3. Option with E[X] = 0.50 assumes a symmetric distribution, but the density is skewed toward zero.
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