average mean variance random variable normal normal distribution independent and identically distributed random variables sequence random variables
The theorem states that the average of many independent and identically distributed random variables with finite variance tends towards a normal distribution irrespective of the distribution followed by the original random variables. Formally, let 




X

1


,

X

2


,
…



{\displaystyle X_{1},X_{2},\dots \,}

 be independent random variables with mean 



μ


{\displaystyle \mu }

 and variance 




σ

2


>
0.



{\displaystyle \sigma ^{2}>0.\,}

 Then the sequence of random variables
