Laws of Large Numbers and Central Limit Theorems

Experiments on Laws of Large Numbers

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

After reviewing these, you will be given the chance to try experimenting interactively.

Experiment 1: samples drawn from a normal distribution.

Exact p.d.f.'s

Empirical (Monte-Carlo) p.d.f.'s

Experiment 2: we draw samples from an exponential distribution which is positively skewed.

Exact p.d.f.'s

Empirical (Monte-Carlo) p.d.f.'s

Experiment 3: we work with a binomial distribution and the sample proportion of successes.

Exact p.d.f.'s

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

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.

Exact p.d.f.'s

Empirical (Monte-Carlo) p.d.f.'s

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).
- THETA=3 --- Finite Mean, Finite Variance.
  
  Empirical (Monte-Carlo) p.d.f.'s
- THETA=1.5 --- Finite Mean, Infinite Variance.
  
  Empirical (Monte-Carlo) p.d.f.'s
- THETA=0.5 --- Infinite Mean (& Inifinite Variance).
  Empirical (Monte-Carlo) p.d.f.'s