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IAI Actuarial Core Principles · Actuarial Statistics · Credibility theory: Bayesian and empirical Bayes credibility premiums

Claim counts per year are Poisson(λ) with prior λ ~ Gamma(α = 3, rate β = 6). After n years of experience with sample mean x̄, the posterior mean is written Z·x̄ + (1 − Z)·(prior mean). What is Z when n = 4?

Z equals n/(β + n) = 4/10 = 0.4. The posterior mean (3 + 4x̄)/10 splits into 0.4 times the sample mean plus 0.6 times the prior mean of 0.5, so the credibility factor is 0.4.

  1. A0.250
  2. B0.333
  3. C0.400
  4. D0.571Correct
  5. 0.667

Explanation

Posterior mean = (α + n x̄)/(β + n). So Z = n/(β + n) = 4/(6 + 4) = 0.4. Check: (3 + 4x̄)/10 = 0.4x̄ + 0.3 = 0.4x̄ + 0.6·0.5, consistent with prior mean 0.5. The value 0.571 would arise from 4/7, using the wrong β.

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