IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables
Given Θ=θ, N is Poisson with mean θ. Θ is exponential with mean 2. Which is the value of Var(N)?
Var(N) is 6. By the conditional variance formula, the expected conditional variance is E(Θ)=2 and the variance of the conditional mean is Var(Θ)=4 for an exponential with mean 2, giving 6. Taking only 2 ignores parameter uncertainty.
- A2
- B4
- C6Correct
- D8
- 10
Explanation
E(N)=E(Θ)=2. Var(N)=E[Var(N|Θ)]+Var[E(N|Θ)]=E(Θ)+Var(Θ)=2+4=6. The value 2 is wrong because it omits the variance of the mean, and 4 uses only Var(Θ).
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