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IAI Actuarial Core Principles · Actuarial Statistics · Central limit theorem

Let X1, X2, ... be independent Bernoulli variables with P(Xi=1)=0.2. Using the CLT with a continuity correction, what is the approximate probability that the sum of 100 of them is at most 25? (Φ(1.0)=0.8413, Φ(1.125)=0.8697, Φ(1.25)=0.8944, Φ(1.5)=0.9332, Φ(1.375)=0.9154)

The sum is approximately normal with mean 20 and standard deviation 4. With continuity correction the boundary is 25.5, giving z = 1.375 and a probability of about 0.9154.

  1. A0.8413
  2. B0.8697Correct
  3. C0.8944
  4. D0.9154
  5. 0.9332

Explanation

The sum is Binomial(100,0.2) with mean 20 and variance 16, so sd 4. With continuity correction P(S ≤ 25) ≈ P(Z ≤ (25.5 − 20)/4) = Φ(1.375) = 0.9154. Check: the options list Φ(1.375)=0.9154 as option D. Correcting: the answer is 0.9154, not 0.8697.

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