# Philosophical question about central limit theorem

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

Let Xi 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 ... Xn)/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 $\sqrt{n}$. The quantity that is approximately normally distributed (for a sample of size n , sample mean $S_n$ and population mean $\mu$ for each individual random variable) is $\sqrt{n}(S_n - \mu)$.

Last edited:
chiro