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IAI Actuarial Core Principles · Actuarial Statistics · Central limit theorem

The daily claim amounts at an Indian insurer's branch are independent with mean Rs 5,000 and standard deviation Rs 2,000. Using the central limit theorem, what is the approximate probability that the total of 100 such claims exceeds Rs 5,40,000? (Phi(2) = 0.9772, Phi(1.5) = 0.9332, Phi(1) = 0.8413)

The probability is about 0.0228. The total has mean Rs 5,00,000 and standard deviation Rs 20,000, so Rs 5,40,000 is 2 standard deviations above the mean, and the upper normal tail beyond 2 is 1 minus 0.9772.

  1. A0.0228Correct
  2. B0.0668
  3. C0.1587
  4. D0.9772
  5. 0.8413

Explanation

Total has mean 100 x 5000 = 5,00,000 and standard deviation 2000 x 10 = 20,000. Z = (5,40,000 - 5,00,000)/20,000 = 2. P(Z>2) = 1 - 0.9772 = 0.0228. The value 0.9772 is the lower tail, and 0.1587 comes from wrongly using Z=1.

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