IAI Actuarial Core Principles · Actuarial Statistics · Expectations and conditional expectations
A claim amount X (in rupees thousand) has density f(x) = 2/x^3 for x > 1 and 0 otherwise. What is E[X]?
E[X] equals 2. Integrating x times 2/x^3 gives 2 times the integral of x^-2 from 1 to infinity, which equals 2 times 1. The mean is therefore finite and equal to 2 thousand rupees.
- A1
- B1.5
- C2Correct
- D4
- Infinite
Explanation
E[X] = ∫ from 1 to ∞ of x(2/x^3) dx = 2∫ x^-2 dx = 2[−1/x] from 1 to ∞ = 2. The density integrates to 1, so it is valid. The answer 1 comes from forgetting the factor 2.
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