Explain Central Limit Theorem with random variables

Question

Explain Central Limit Theorem with an example of random variables.

Answer 1

Assume that X1, X2, ... , X_n be a sequence of random variables that are independent and equally distributed (i.i.d.), along with finite mean, m and s that is standard deviation.

The sum has mean mn and standard deviation as s√n.

By the Central Limit Theorem n will be larger as the distribution of Sn tends to the normal distribution. More exactly, the distribution of S‾n converges to the normal distribution along with zero mean and unit standard deviation when n tends to infinity, if we work with the scaled quantity. The cumulative distribution for S‾n approaches as for the standardized normal distribution.

S‾n = (Sn – mn)/(s √n)