Suppose I have a simple uniform continuous "unit" distribution X:
$\begin{align*} \forall y \in \mathbb{R} \implies \\ y < 0 : & P(X < y) = 0 \\ y \in [0,1] : & P(X < y) = y \\ y > 1 : & P(X < y) = 1 \\ \end{align*}$
Let $Y_n$ be a random variable equal to the mean of $n$ independent variables with a distribution of X.
Let $Z_n$ be a random variable defined as $Y_n$ normalized. That is:
$\begin{align*} Z_n = \frac{Y_n - mean(Y_n)}{stddev(Y_n)} \end{align*}$
Is it correct to say that as $i \rightarrow \infty$, $Z_i$ approaches the standard normal distribution by the central limit thereom ?
If so, then is there some way we can derive/calculate the formula for the normal distribution based on the formula for X above? Perhaps using some calculus or whatever? How would this be done?