Sample Question #82 (probability theory)

Briefly explain the difference between the *Central Limit Theorem *and the *Law of Large Numbers.*

(Comment: these two are very fundamental distribution concepts behind statistical modeling, but a lot of candidates who otherwise know statistics do not know what these theorems say or which is which)

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ANSWER

The Central Limit Theorem says that the sum of a many independent, identically distributed (i.i.d.) random variables is approximately normally distributed. An immediate corollary is that the mean of these random variables is also approximately normally distributed. This theorem makes it convenient to study non-normal samples.

The Law of Large Numbers relates samples from a distribution to the (theoretical) distribution itself: it says that the mean of any sample with a large enough size is very similar to the true mean of the population. A corollary is that the larger the sample size, the closer the sample mean is to the population mean. This theorem allows us to use sample mean to represent the expectation of the distribution.

Both theorems have a "weak" and "strong" versions. The weak versions state the result in terms of convergence in probability, whereas the strong versions state the the probability of the phenomenon approaches 1 as size increases.