IAI Actuarial Core Principles · Actuarial Statistics · Bayesian inference: priors, posteriors, loss functions and credible intervals
Observations x1,...,x_n are normal with unknown mean μ and known variance 16. The prior for μ is N(50, 16). A sample of n = 4 has mean 58. What is the posterior mean of μ?
The posterior mean is a precision-weighted average of the prior mean 50 and the sample mean 58. The data carry weight 4/5 because the sample has four times the prior precision, giving 50 + 0.8×8 = 56.4.
- A52.0
- B54.0
- C55.2Correct
- D56.4
- 58.0
Explanation
Prior precision is 1/16 and data precision is 4/16, so the posterior mean is (50×1 + 58×4)/5 = (50 + 232)/5 = 56.4. Check: weight on data is 4/5 = 0.8, so 50 + 0.8×8 = 56.4. The 54.0 option uses equal weights, which ignores that n = 4.
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