Let X 1 , , X n X 1 , , X n be iid

Let $X_1,\dots ,X_n$ be iid random variables having continuous uniform distributions of the form $f\left(z\right)=I_{(0,1)}\left(z\right)$.

(a) Define an asymptotic distribution for ${\overline{X}}_n=n^{-1}{\sum }_{i=1}^n{X}_i$.

(b) Using your result from (a), argue that $\left({\sum }_{i=1}^{12}{X}_i\right)-6\approx Z\sim N(0,1)$.

(This approximation is very accurate, and is sometimes used for simulating $N(0,1)$ outcomes using a uniform random number generator.)