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IAI Actuarial Core Principles · Actuarial Statistics · Central limit theorem

Policy losses are independent, each Poisson with mean 4. Using the central limit theorem with continuity correction, which expression approximates P(total of 25 policies is at most 110)? (Total is Poisson with mean 100.)

The approximation is Phi((110.5 - 100)/10). The total is Poisson with mean and variance 100, so the standard deviation is 10, and the continuity correction for P(S at most 110) moves the boundary up to 110.5.

  1. APhi((110 - 100)/10)
  2. BPhi((110.5 - 100)/10)Correct
  3. CPhi((109.5 - 100)/10)
  4. DPhi((110.5 - 100)/100)
  5. Phi((110.5 - 4)/10)

Explanation

The total is Poisson with mean 100 and variance 100, so SD is 10. For a discrete P(S<=110) the continuity correction uses 110.5. Phi((110.5-100)/10) = Phi(1.05). Using 109.5 would approximate P(S<=109); dividing by 100 uses variance instead of SD.

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