"Show experimentally" that for large $N$, $X$ appears to be normally distributed.

Question

I'm a bit confused about the following problem:

Let $X$ be the random variable $$X = \frac{X_1+X_2+...+X_N}{\sqrt{N}}$$ where $X_k$ is the outcome from the $kth$ flip of a fair coin where heads yields the value $1$ and tails $-1$. Show that $E[X]=\mu=0$ and $\sigma^2= E[X^2]=1$. But what is its distribution? Show experimentally that for large $N$, $X$ appears to be normally distributed. You will be substantiating the celebrated $\textit{central limit theorem}$ stated in any book on statistics.

Now, I do understand the idea of the CLT. I also understand why mean is zero and variance is one. However, I am unsure what the problem is asking for when it asks me to "show experimentally."

Answer 1

I guess what they mean is that you should simulate values from the distribution of $$ X = \frac{X_1+X_2+...+X_N}{\sqrt{N}}. $$ You can use these to plot a histogram and then confirm that it looks like a bell shaped normal curve.

Answer 2

Without the rest of the context, my guess would be that they want you to flip (or, hopefully, write a Mathematica/Matlab script or other computer program to simulate flipping) a coin $N$ times for some large $N$, and write down the value of $X$ for that sequence (i.e., for $N = 2$, two heads would correspond to $X = (1 + 1)/\sqrt{2} = \sqrt{2}$). Repeat the process a number of times to get a distribution for $X$, then check that it's reasonably close to normal.