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IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties

A claims analyst models the number of claims per policy as Poisson with mean λ. Over 50 policies the total number of claims is 85. What is the maximum likelihood estimate of the probability that a policy has no claims?

The estimate is 0.1827. The MLE of the Poisson mean is 85/50 = 1.7, and by invariance the MLE of P(no claims) is e^(-1.7), which equals 0.1827. The value 0.8173 is the complement, the chance of at least one claim.

  1. A0.1827Correct
  2. B0.3000
  3. C0.7000
  4. D1.7000
  5. 0.8173

Explanation

The MLE of λ is the sample mean, 85/50 = 1.7. By invariance of MLEs, the MLE of P(X=0) = e^(-λ) is e^(-1.7) = 0.1827. The value 0.8173 is the probability of at least one claim, and 1.7 is just λ-hat.

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