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IAI Actuarial Core Principles · Actuarial Statistics · Central limit theorem

The claim amounts on a motor policy are independent with mean ₹8,000 and standard deviation ₹6,000. Using the CLT, what is the approximate probability that the total of 100 claims exceeds ₹830,000? (Φ(0.5)=0.6915, Φ(1)=0.8413, Φ(1.5)=0.9332, Φ(2)=0.9772)

The total of 100 claims is approximately normal with mean ₹800,000 and standard deviation ₹60,000. The z-value for ₹830,000 is 0.5, so the upper-tail probability is 1 − 0.6915, which is about 0.3085.

  1. A0.0668
  2. B0.1587
  3. C0.3085Correct
  4. D0.0228
  5. 0.4013

Explanation

Total S has mean 100×8,000 = 800,000 and standard deviation 6,000×√100 = 60,000. Then z = (830,000 − 800,000)/60,000 = 0.5, so P(S > 830,000) = 1 − 0.6915 = 0.3085. Using 6,000 as the sd of the total (z=5) or dividing wrongly would give other values; 0.1587 would come from z=1.

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