IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables
Given Λ = λ, the claim count N is Poisson(λ). Λ has density f(λ) = e^(-λ) for λ > 0 (exponential with mean 1). What is P(N = 0)?
P(N = 0) is 1/2, found by integrating the Poisson probability e^(-λ) against the exponential density e^(-λ) over all λ.
- A1/4
- B1/3
- C1/2Correct
- D2/3
- e^(-1)
Explanation
P(N=0) = integral of e^(-λ) * e^(-λ) dλ over (0,∞) = integral of e^(-2λ) dλ = 1/2. The option e^(-1) wrongly substitutes the mean λ = 1 instead of integrating over the mixing distribution.
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