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IAI Actuarial Core Principles · Actuarial Statistics · Central limit theorem

Which of the following is a situation in which the standard CLT for the sample mean of iid variables does NOT apply?

The standard CLT does not apply to Cauchy observations because the Cauchy distribution has no finite mean or variance. The sample mean of Cauchy variables is itself standard Cauchy, so it never becomes normal. The other listed distributions all have finite variances.

  1. AObservations are Poisson with mean 3
  2. BObservations are uniform on (0,1)
  3. CObservations are standard CauchyCorrect
  4. DObservations are exponential with mean 10
  5. Observations are Bernoulli with p = 0.3

Explanation

The CLT requires finite mean and variance. The Cauchy distribution has no mean or variance, and the mean of n Cauchy variables is again Cauchy. The other distributions all have finite variances.

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