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IAI Actuarial Core Principles · Actuarial Statistics · Central limit theorem

An insurer has 400 independent policies. Each policy's annual claim amount has mean Rs 5,000 and standard deviation Rs 12,000. Using the central limit theorem, what is the approximate probability that total annual claims exceed Rs 2,240,000? (Φ(1.0)=0.8413, Φ(0.5)=0.6915, Φ(1.5)=0.9332, Φ(2.0)=0.9772.)

The probability is about 0.1587. Total claims have mean Rs 2,000,000 and standard deviation Rs 240,000, so Rs 2,240,000 is one standard deviation above the mean, and the upper tail probability under the normal approximation is 1 − 0.8413.

  1. A0.0668
  2. B0.3085
  3. C0.1587Correct
  4. D0.0228
  5. 0.8413

Explanation

Total mean = 400×5,000 = 2,000,000. Total SD = 12,000×20 = 240,000. Z = 240,000/240,000 = 1, so P(exceed) = 1 − 0.8413 = 0.1587. Using 0.8413 ignores the 'exceed' direction.

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