IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables
X is the number of claims in a year, with P(X=0)=0.6 and P(X=1)=0.4. Given X=0 the total claim cost Y is zero. Given X=1, Y is exponential with mean Rs 50,000. What is E(Y)?
E(Y) is Rs 20,000. By the tower property, E(Y) is the weighted average of conditional means: 0.6 times zero plus 0.4 times Rs 50,000. Using Rs 50,000 ignores that no claim occurs with probability 0.6.
- ARs 20,000Correct
- BRs 30,000
- CRs 50,000
- DRs 25,000
- Rs 40,000
Explanation
E(Y)=E[E(Y|X)]=0.6×0+0.4×50,000=20,000. Rs 50,000 is wrong because it ignores the chance of no claim, and Rs 30,000 uses 0.6 as the weight on the exponential component.
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