IAI Actuarial Core Principles · Actuarial Statistics · Credibility theory: Bayesian and empirical Bayes credibility premiums
Claim numbers for a risk in a year are Poisson with mean Λ. The prior for Λ is Gamma with shape α = 3 and rate β = 2 (mean 1.5). The risk produced 4 claims in one year. Under squared error loss, what is the Bayesian estimate of Λ for next year?
The estimate is 7/3, about 2.33. The Gamma(3, 2) prior updates with 4 claims in 1 year to Gamma(7, 3), and the Bayesian premium under squared error loss is the posterior mean, 7 divided by 3.
- A1.5
- B2.0Correct
- C2.33
- D3.5
- 1.75
Explanation
The posterior is Gamma(α + x, β + n) = Gamma(3 + 4, 2 + 1) = Gamma(7, 3). The posterior mean is 7/3 = 2.33. Option 2.0 is wrong because it uses the wrong update; 3.5 is 7/2, which forgets to add the exposure to the rate.
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