Central Limit Theorem¶
The basic ida that if we have i.i.d random variables \(X_1, \cdots, X_n\) where each of those random variables \(E[X_i] = \mu, var(X_i) = \sigma\). Than if we take a the sum of those n i.i.d r.v’s \(S_n = X_1 + \cdots X_n\) than we get a new distribution where:
\[E[S_n] = n E[X_i] \]
\[ var(S_n) = n \sigma^2 \]
We can visualize this as a flat distribution.
Now we can push this futher to standardise this distribution to get a new distribution:
\[ Z_n = \frac{S_n - n \mu}{ \sqrt{n}\sigma}\]
This distribution now has:
\[\begin{split}
E[Z_n] = 0 \\
var(Z_n) = 1
\end{split}\]
and if \(n \rightarrow \infty\) than the CDF of \(Z_n\) converges to the standard normal CDF. Or more simply \(Z_n \sim N(0,1)\)