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IAI Actuarial Core Principles · Actuarial Statistics · Central limit theorem

The sample mean X̄ of n independent observations from a skewed distribution with mean μ and variance σ² is approximately normal for large n. Which statement about this result is correct?

By the central limit theorem, the sample mean of n independent identically distributed observations is approximately normal with mean mu and variance sigma squared divided by n. The version with mean n mu and variance n sigma squared describes the sum instead.

  1. AX̄ is approximately N(μ, σ²/n)Correct
  2. BX̄ is approximately N(μ, σ²)
  3. CX̄ is approximately N(nμ, nσ²)
  4. DX̄ is exactly normal for any n
  5. X̄ is approximately N(μ, σ/n)

Explanation

By the CLT, the sample mean has mean μ and variance σ²/n. N(nμ, nσ²) describes the sum, not the mean. Exact normality holds only if the parent is normal.

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