Deviation from the Mean · Central Limit Theorem (Optional)
Lesson 1
Nikolai Chukhin · Alexander S. Kulikov
Histograms of many real-world datasets have a bell shape: student grades, infant weights, human heights, and tree leaf lengths.
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This bell shape resembles the graph of the function \(e^{-x^2/2}\) (appropriately shifted and stretched). =0/image1.png)
For the curious 🤓
A distribution whose density function forms such a bell shape is known as normal and is one of the most important continuous distributions. A normal distribution with parameters \(\mu \in \mathbb{R}\) and \(\sigma>0\) is given by the following density function: \[F(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2} \left(\frac{x-\mu}{\sigma}\right)^{2}\right) \ .\] The expectation of such a distribution is \(\mu\), and the variance is \(\sigma^{2}\).
Central limit theorems (CLT) describe this effect in probability theory: as \(n\) increases, the distribution of the mean of \(n\) independent identically distributed random variables tends to a normal distribution. We will prove a simple special case of the CLT below.