Does the central limit theorem apply for random variables with densities which are not asymptotic?

Question

I’m probably missing something, but how can for example the mean of samples of nutritional needs (assuming i.i.d. random variables) be normally distributed when there is a certain minimum in the original density function below which it is zero? (There is for example certainly nobody who requires only $500 \; kcal$ per day so the probability to get $\bar x =500$ is zero while the approximation of the distribution of the mean with a bell curve would be strictly greater than zero for $500 \; kcal$.)

Answer 1

The probability that a (nondegenerate) normal random variable is $500$ is zero as well. What the CLT asserts is that the probability that the renormalized empirical mean lies in some given interval converges to the probability that some normal random variable lies in the same interval.

You might be missing the renormalizing part as well hence let us recall that if $\bar X_n$ denotes the empirical mean, the convergence in distribution in the CLT is not concerned with $\bar X_n$, which simply converges to the common mean $\mu$, but with $\sqrt{n}(\bar X_n-\mu)$.

Answer 2

Well, according to CLT, if n -> infinity, the distribution converges to normal distribution with mean m and variance sigma(finite). You are probably guessing it converges to standard normal distribution with mean 0 and variance 1, which it does, if the variables in consideration are standardized(i.e. subtract the mean and divide by standard deviation for each sample value). Hope this clarifies your confusion! :)