Let be an infinite sequence of independent and identically distributed random variables, each with the same mean and standard deviation , and let
Which of these statements is true?

Question

Let    be an infinite sequence of independent and identically distributed random variables, each with the same mean    and standard deviation   , and let   
 Which of these statements is true?

Accepted Answer

The Law of Large Numbers and Central Limit Theorem both try to approximately tell us the behaviour of the sample mean   .

Accepted Answer

The Central Limit Theorem gives us the approximate shape of the distribution of   .

Accepted Answer

Central Limit Theorem is not restricted by the underlying distribution of   -s.

Answer

Central Limit Theorem only holds if each    is normally distributed (i.e.    ).

Answer

If the sample size    is small (e.g.   ), then the Law of Large Numbers on    does not hold, but the Central Limit Theorem on    is still valid (i.e.    if   ).

Crowdly

Let be an infinite sequence of independent and identically distributed random...

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