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IAI Actuarial Core Principles · Actuarial Statistics · Central limit theorem

X ~ Binomial(100, 0.4). Using a normal approximation with continuity correction, which expression approximates P(X ≤ 35)? Here sd = √24 ≈ 4.899.

P(X ≤ 35) is approximated by the normal probability below 35.5, so it equals Φ((35.5−40)/4.899). The continuity correction adds 0.5 to the upper limit; using 34.5 would instead estimate P(X ≤ 34).

  1. AΦ((35−40)/4.899)
  2. BΦ((35.5−40)/4.899)Correct
  3. CΦ((34.5−40)/4.899)
  4. DΦ((36−40)/4.899)
  5. 1−Φ((35.5−40)/4.899)

Explanation

For a discrete X, P(X ≤ 35) corresponds to the continuous region below 35.5. Standardising with mean 40 and sd √24 gives Φ((35.5−40)/4.899). Using 34.5 would approximate P(X ≤ 34) instead.

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