IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables
Given Θ = θ, the claim count N is Poisson(θ). Θ has a Gamma distribution with mean 2 and variance 1. What is Var(N)?
By the law of total variance, Var(N) = E[Θ] + Var(Θ) = 2 + 1 = 3. The mixing over the Gamma parameter adds extra variance beyond the Poisson value of 2.
- A1
- B2
- C3Correct
- D4
- 5
Explanation
E[Var(N|Θ)] = E[Θ] = 2. Var(E[N|Θ]) = Var(Θ) = 1. Total variance = 2 + 1 = 3. Answer 2 ignores the variability of Θ.
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