1. ## Central Limit Theorem

From my understanding, according to the central limit, $T = X_1 + X_2 + ... + X_n$ should behave (roughly) like an N(0,1) distribution for a large enough n.

I'm trying to show this by simulation. I created 1000 $X_i$ iid ~U[0,1]. So according to CLT, T~N(0,1). But how would I show this?

2. According to Central Limit Theorem, $\frac{T-mean(T)}{sd(T)}$ follows a $N(0,1)$ distribution for large n. Also, if $X_i$'s are iid $U(0,1)$ random variables, then their sum do not follow $N(0,1)$ distribution.

3. the average of uniforms, when standardized should be approximately normal.
same goes for the sum

4. so $\frac{T - \frac{n}{2}}{n*\sqrt{\frac{1}{12}}}$ is approximetaly N(0,1)?

5. since these are iid with a second moment we have

${\bar X-{1\over 2}\over \sqrt{1\over 12n}}\to N(0,1)$

we have ${n\bar X-{n\over 2}\over n\sqrt{1\over 12n}}\to N(0,1)$

6. So if I were to plot that, I would get a rouch bell curve, correct? But I try to do this on R

for (i in 1:1000){
+ T=(sum(runif(1000))-(1000/2))/1000*sqrt(1/12*1000)
+ V[i]=T
+ }

I create runif(1000) creates 1000 U[0,1] and T is me trying to standardize it using CLT. I do this 1000 times, but I dont get a bell curve when I plot V.

7. Did you try....... ${T_n-{n\over 2}\over \sqrt{n\over 12}}$

because yours is wrong, that n in the denominator is incorrect.