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FRM Part I · FRM Exam Part I · Common Univariate Random Variables

Which statement about the Central Limit Theorem is most accurate for i.i.d. observations with finite variance?

For i.i.d. observations with finite variance, the standardized sample mean approaches a standard normal distribution as the sample size grows, whatever the population shape. The individual observations do not become normal, and the standard error shrinks rather than rises as n increases.

  1. AThe distribution of individual observations becomes normal as the sample size grows
  2. BThe distribution of the standardized sample mean approaches a standard normal as the sample size growsCorrect
  3. CThe sample mean is normal only if the underlying population is normal
  4. DThe standard error of the sample mean increases with the sample size

Explanation

The CLT concerns the sampling distribution of the (standardized) sample mean, which approaches normal for large n regardless of the population shape. The raw observations keep their own distribution, and the standard error falls with sqrt(n).

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