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IAI Actuarial Core Principles · Actuarial Statistics · Expectations and conditional expectations

The number of claims N in a year has mean 4. Given N, each claim amount is independent with mean Rs 5,000. Using the law of total expectation, what is the expected total annual claim cost S?

The expected total is Rs 20,000. Conditional on N claims, the expected total is 5,000 times N, and taking the expectation over N gives 5,000 times the mean of 4, so the product of mean claim count and mean claim size applies.

  1. ARs 5,000
  2. BRs 9,000
  3. CRs 20,000Correct
  4. DRs 25,000
  5. Rs 1,250

Explanation

E[S] = E[E[S|N]] = E[N × 5,000] = 4 × 5,000 = Rs 20,000. Adding the two means (Rs 9,000) is the wrong operation, since the cost is a sum of N claims and not a sum of the two means.

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