Experiments on Central Limit Theorems

The results of five sets of experiments are overviewed here, characterized by five true underlying distributions -- populations.

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• Experiment 1: samples drawn from a normal distribution.
  
  Empirical (Monte-Carlo) p.d.f.'s
  
  


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• Experiment 2: we draw samples from an exponential distribution which is positively skewed. We observe that the CLT still works for large sample sizes and produces an approximately bell-shaped distribution for the standardized mean, despite the underlying skewness.
  
  Empirical (Monte-Carlo) p.d.f.'s
  
  

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• Experiment 3: we work with a binomial distribution and the sample proportion of successes. The CLT case then illustrates the De Moivre --- Laplace theorem. Observe that the ``gaps" due to the underlying discreteness are filled as the sample size grows --- in the limit the distribution becomes normal.
  
  Empirical (Monte-Carlo) p.d.f.'s
  
  Missing File

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• The last two sets of experiments highlight the importance of the necessary assumptions for the LLN's and CLT's to work.

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  • Experiment 4: Here the observations are drawn from a Cauchy distribution, which has no finite moments. As a result, both the LLN and the CLT fail.
    
    Empirical (Monte-Carlo) p.d.f.'s
    
    
    

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  • Experiment 5:Here the Pareto distribution is used to generate the draws. A single parameter of this distribution determines whether the variance will be infinite (in which case the CLT fails to hold) and/or the mean will be infinite (leading to a failure of the LLN as well).
    

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    • THETA=3 --- Finite Mean, Finite Variance.
      Empirical (Monte-Carlo) p.d.f.'s
      
      

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      THETA=1.5 --- Finite Mean, Infinite Variance.
      Empirical (Monte-Carlo) p.d.f.'s
      
      

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    • THETA=0.5 --- Infinite Mean (& Infinite Variance).
      Empirical (Monte-Carlo) p.d.f.'s
      
      

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