This simple project is designed to observe the implication of Central Limit Theorem on probability distributions. The project consists of several experiments on sums of independent and identically distributed random variables (IID).
The Central Limit Theorem is stated as follows:
Given , a sequence of IID random variables with expected value
and variance
, the CDF of
has the property
Briefly, the theorem implies that: As n increases, the sum of n IID random variables converges to a Gaussian Distribution.
Following four experiments with four different probability functions designed to show the sums converges to Gaussian indeed. The probability functions used in these experiments specified below:
| The Experiment Number | The Type of The Individual RV |
|---|---|
| 1st experiment | Uniform R.V. |
| 2nd experiment | Exponential R.V |
| 3rd experiment | Bernoulli R.V |
| 4th experiment | Poisson R.V. |
For all experiments, 1000 samples of random sums are generated for corresponding n values. Then, they are normalized and shown on a histogram. n values used in these experiments are chosen 1, 2, 5, 10, 20, 40 for demonstration purposes. The resulting histograms are plotted with a Gaussian probability distribution function (PDF) or Gaussian cumulative distribution function (CDF) according to the context of the experiment.
Scripts of the experiments are given in the repository.
The results of the experiments are below. As n increases, the sums are observed to converge to the corresponding Gaussian CDF/ PDF.
1000 samples of sum of n=[1 2 5 10 20 40] IID Exponential R.V. are put to historgrams plotted below.
1000 samples of sum of n=[1 2 5 10 20 40] IID Bernoulli (with p=0.5) R.V. are put to historgrams plotted below.
[1] Yates, R. D. & Goodman, D. J. (2013), Probability and Stochastic Processes - a Friendly Introduction for Electrical and Computer Engineers [2nd ed.], New Jersey: WILEY