Philosophical question about central limit theorem

[coquelicot]

Well, this is probably a stupid question, but I don't see why (yet).

Let X_i be random variables identically distributed, with mean 0, such that the cumulative distribution is = 0 for all -1 < x < 1. So, I believe it is clear that for all n, the cumulative distribution of Z = (X₁ + X₂ ... X_n)/n is = 0 for all x < -1. But the central limit theorem implies that this distribution (normalized by the square root of n), converges in distribution to the Normal distribution. So, for some n sufficiently large, the cumulative distribution of Z must be > 0 for some x < -1. Where is the fallacy in this paradox?
thx.

 

[Stephen Tashi]

But the central limit theorem implies that this distribution (normalized by the square root of n),
.

The normalization involves multiplying the sample mean by [itex] \sqrt{n} [/itex]. The quantity that is approximately normally distributed (for a sample of size n , sample mean [itex] S_n [/itex] and population mean [itex] \mu [/itex] for each individual random variable) is [itex] \sqrt{n}(S_n - \mu) [/itex].

 

[chiro]

If you have issues with the application of the theorem to a particular distribution I would recommend opening a statistical software package and simulating a number of random variables and then graphing the result of the random vectors that get the arithmetic mean and see what happens.

It should help convince you through some examples of how the CLT holds.

 

[coquelicot]

Thx all.
I have understood my error.

 

[blue_raver22]

I appreciate your curiosity and willingness to question assumptions. The central limit theorem is a powerful tool in statistics that allows us to make predictions about the behavior of a large number of random variables. However, it is important to understand the assumptions and limitations of this theorem.

In this case, the fallacy lies in assuming that the cumulative distribution of Z will always be equal to 0 for all x < -1. This is not necessarily true, as the central limit theorem only guarantees that the distribution of Z will approach the Normal distribution as n becomes large. This means that for smaller values of n, the distribution of Z may not follow the Normal distribution and may have non-zero values for x < -1.

Additionally, the assumption that the random variables Xi are identically distributed with mean 0 may also be a limitation. The central limit theorem requires that the random variables have finite variance, so if the variance of Xi is very small, it may take a larger value of n for the distribution of Z to approach the Normal distribution.

In conclusion, while the central limit theorem is a powerful tool, it is important to understand its assumptions and limitations. It is always good to question and explore paradoxes, but it is also important to carefully examine the underlying assumptions and data before drawing conclusions.