Central limit theorem

If you have studied probability theory then you would have definitely learned about central limit theorem. This theorem is used to state those conditions in which mean of all the random and independent variables are normally distributed. According to this theorem, all the random variables should be distributed identically. If you go through the quantities of real world then you will find that all the unobserved and random events are balanced. Through this theorem you can get the partial explanation of the probability of normal distribution and its prevalence as well. This theory will also be justifying the statistics of large sample approximation in all the controlled experiments.

If you want to understand this theorem then you can easily understand through rolling dice. Sum distribution or average distribution of all the rolled numbers are approximated through normal distribution. You can determine the parameters of the same empirically. Probability theory defines this theorem as a weak convergence set of theories. This is for expressing that sum of all the random and independent variables will be distributed according to the normal distributions.

You will get many interesting and useful examples as well as applications that arise from this theorem. This theorem is also used for the explanation of bell curve appearance. You can also apply this theorem to real world. Most of the mathematicians and researchers make use of this theorem. This theory is having a very interesting history. First version of this central limit theorem was given by a mathematician Abraham Moivre de who was a French mathematician. This mathematician made use of the normal distribution to explain this theory and had explained through the example of tossing coins. This theory got published for the first time in an article in year 1733. It was in year 1812 that this theory got published again by a French mathematician Pierre Laplace Simon. This time this theory was explained through binomial distribution. But both of the time this theory did not receive any attention. It was in year 1901 that another attempt was made by a mathematician to explain this theory again.

This time this theory was explained by a Russian mathematician Lvapunov Aleksandr. This mathematician was successful in explaining the how this theory works and how you can make use of this theory in general. Nowadays, this theorem is used as unofficial sovereign of the probability theory. This theorem will be defining characteristics of all population mean samples. This has been created through infinite number of population samples which are of the size N. all of them are also drawn from the parent population. According to this theorem population mean will always be equivalent to the parent population mean through which population samples were drawn. According to this theorem population standard deviation will also be equivalent to the parent population standard deviation. Another very important point that you need to keep in your mind is that mean distribution will always be increasing in approximation to the normal distribution and this is of the size N.