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IAI Actuarial Core Principles · Actuarial Statistics · Central limit theorem

X ~ Binomial(40, 0.5) is approximated by a normal distribution with continuity correction. Which expression approximates P(X ≤ 24)?

P(X ≤ 24) is approximated by P(Z ≤ (24.5 − 20)/√10). The binomial has mean 20 and variance 10, and the continuity correction adds 0.5 to the upper boundary so the whole probability block at 24 is included.

  1. AP(Z ≤ (23.5 − 20)/√10)
  2. BP(Z ≤ (24 − 20)/√10)
  3. CP(Z ≤ (24.5 − 20)/√10)Correct
  4. DP(Z ≤ (25 − 20)/√10)
  5. P(Z ≤ (24.5 − 20)/√20)

Explanation

Mean = 20 and variance = 40×0.5×0.5 = 10. For a discrete X, P(X ≤ 24) is treated as P(Y < 24.5) under the normal approximation. So the standardised value is (24.5 − 20)/√10. Using 23.5 would be P(X ≤ 23), and using √20 uses the wrong standard deviation.

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