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

The number of claims N in a year for a policyholder has mean 3. Each claim amount X has mean Rs 20,000, and claim amounts are independent of N and of each other. Using the law of total expectation, what is the expected total annual claim S = X1 + ... + XN?

The expected total claim is Rs 60,000. By the law of total expectation, E[S] equals E[N] times E[X], which is 3 multiplied by Rs 20,000, because claim sizes are independent of the number of claims.

  1. ARs 23,000
  2. BRs 60,000Correct
  3. CRs 6,667
  4. DRs 20,000
  5. Rs 120,000

Explanation

E[S] = E[E[S|N]] = E[N·E[X]] = 3 × 20,000 = 60,000. Adding the means (Rs 23,000) mixes a count with a monetary amount. Dividing gives Rs 6,667, which is meaningless here.

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