7.2 Central Limit Theorem

The Central Limit Theorem tells us that the average of an i.i.d. sample of a random variable \(X\) will converge in distribution to the Normal \[z_n = \frac{\bar{x}_n-\mu}{\sigma / \sqrt{n}} \Rightarrow F_{z_n}(a) = \Phi(a)\] where \[\bar{x}=\frac{1}{n} \sum_{i=1}^{n} x_i\] The Central Limit Theorem is key for what we are going to see in the remaining sections. What it states is that no matter the underlying distribution (discrete or continuous), if we sample repeatedly and write down the mean of the sample \(i\), i.e., \(\bar{x}_i\), those means \(\bar{x}_i\) will be normally distributed.

Figure 7.3: Illustration of the Central Limit Theorem