- Before defining central limit theorem let's clarify the term i.i.d.
- independent and identically distributed (i.i.d) random variables - this means that the random variables we consider have the same probability distribution as the others and they are mutually independent. An example would be two dice experiments performed simultaneously.
- Central limit theorem states that if you take the average of the outcomes of sufficiently large i.i.d random variables, with finite mean and variance, it follows a normal or Gaussian distribution.
- Here is an example. Suppose we throw a dice 10000 times. Then the outcome would more or less look like this (uniform distribution).
- Now, let's throw 2 dices together 10000 times and observe the average (or you can observe the sum of the outcomes for that matter) of outcomes.
- Now, you get the idea. Suppose we throw 50 dices together 10000 times and observe the average (or sum) of outcomes.
- This looks like a normal distribution to me.
- Wikipaedia says: "Since real-world quantities are often the balanced sum of many unobserved random events, this theorem provides a partial explanation for the prevalence of the normal probability distribution".
