#### The Central Limit Theorem (Not Required)

- The observation in 4.3 Approximating Sample Proportion Distribution is a well-known effect. Statisticians has investigated such phenomenon and came up with a theorem for it, known as the Central Limit Theorem.

- The Central Limit Theorem (CLT) states that, in many situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed.

- To explain in math, suppose that X=\frac{1}{n}\left(X_{1}+X_{2}+X_{3}+\ldots+X_{n}\right), then we can assume X \sim \color{red}N\color{black}\left(\mu, \frac{\sigma^{2}}{n}\right). The most important part here is N. as for the mean of variance of X, it can be calculated without the usage of any approximation/assumption (shown below).

- We can find the mean and variance of X as below: