IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties
For a sample of size n from a Poisson distribution with mean λ, the maximum likelihood estimate of P(X = 0) is required. With n = 50 and a sample total of 80, what is the estimate?
The estimate is e^(-1.6) ≈ 0.2019. The MLE of λ is the sample mean 1.6, and by the invariance property the MLE of P(X=0)=e^(-λ) is obtained by substituting it.
- Ae^(-1.6) ≈ 0.2019Correct
- Be^(-0.8) ≈ 0.4493
- C1.6e^(-1.6) ≈ 0.3230
- De^(-80) ≈ 0
- 1 − e^(-1.6) ≈ 0.7981
Explanation
The MLE of λ is x̄ = 80/50 = 1.6. By invariance, the MLE of P(X=0) = e^(-λ) is e^(-1.6) ≈ 0.2019. The 0.3230 option is P(X=1), and 0.7981 is P(X≥1).
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