Help finding central limit theorem approximations - Normal Distribution Equation

Question

I was given f(x)=|x| as a probability distribution. I've summed the results of a Monte Carlo with N terms and plotted a thousand of these sums in a normalized histogram. Now I need to compare this with a normal distribution obtained via the central limit theorem. I've determined the mean will always be 0, and was told the variance for f(x) is 1/2. I am not sure how my normal distributions will depend on N however.

Answer 1

If $X_1, X_2, \dots, X_n$ are independent, identically distributed random variables with $E(X_i) = 0$ and $Var(X_i) = 1/2.$ Then $S_n = \sum_{i=1}^n X_i$ has $E(S_n) = 0$ and $Var(S_n) = n/2.$

By the Central Limit theorem, for sufficiently large $n$, the distribution of $S_n$ will be nearly normal. For the distribution $f_X(x) = |x|, -1 < x < 1,$ the convergence to normal is quite fast, giving a slightly imperfect, but decent fit to normal for $n$ as small as ten.

As an illustration of the speed of convergence consider the following histogram of a simulation in R with $n = 10$ and $m = 10^5$ sums $S_{10}$ (for a histogram more accurately representing the simulated distribution of $S_{10}$ than with your 1000.) Here the relevant normal distribution (red density) is $\mathsf{Norm}(\mu = 0, \sigma=\sqrt{5}).$ [It would be extremely difficult to distinguish data from the distribution of $S_{20}$ from normal, even with formal tests of normality.]

Note: The $X_i$ were simulated as a 50-50 mixture of $\mathsf{Beta}(2,1)$ and its (negative) mirror image.