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IAI Actuarial Core Principles · Actuarial Statistics · Central limit theorem

Let X1, X2, ..., Xn be independent and identically distributed random variables with finite mean μ and finite variance σ² > 0, and let X̄ be their sample mean. Which statement is the standard Central Limit Theorem result?

The Central Limit Theorem says that for iid variables with finite mean and variance, the sample mean standardised by subtracting μ and dividing by σ/√n tends in distribution to a standard normal as n grows, because σ/√n is the standard deviation of the mean.

  1. A(X̄ − μ)/(σ/√n) converges in distribution to a standard normal as n → ∞Correct
  2. B(X̄ − μ)/σ converges in distribution to a standard normal as n → ∞
  3. CX̄ converges in distribution to N(μ, σ²) as n → ∞
  4. D(X̄ − μ)/(σ/n) converges in distribution to a standard normal as n → ∞
  5. X̄ has exactly a normal distribution for every n, whatever the distribution of Xi

Explanation

The CLT states that the standardised mean, (X̄ − μ)/(σ/√n), tends in distribution to N(0,1). The standard deviation of X̄ is σ/√n, not σ or σ/n. The limit is a statement about large n, not exactness for every n.

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