IAI Actuarial Core Principles · Actuarial Statistics · Credibility theory: Bayesian and empirical Bayes credibility premiums
Claim counts for a policyholder in one year are Poisson with mean λ. The prior for λ is Gamma with shape α = 3 and rate β = 6. The policyholder makes 4 claims in 2 years. What is the posterior mean of λ?
The posterior is Gamma with shape 3+4=7 and rate 6+2=8, so the posterior mean is 7/8 = 0.875. The prior parameters are updated by adding total claims to the shape and the number of years to the rate.
- A0.500
- B0.875Correct
- C0.700
- D0.438
- 0.625
Explanation
Poisson-Gamma conjugacy: posterior shape = 3 + 4 = 7 and posterior rate = 6 + 2 = 8. The posterior mean is 7/8 = 0.875. The prior mean 0.5 ignores the data, and 7/16 = 0.438 wrongly adds 2 to the rate twice.
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