Chapter 2 Introduction to statistical inference

$Histograms of $\sqrt{n}(\bar{X}_n-\mu)$ for independent rv’s $X_1,\ldots,X_n$ with expectation $\mu$ and variance $\sigma^2.$ The pdf of $\mathcal{N}(0,\sigma^2)$ is superimposed in red. A clear convergence appears, despite $X_1,\ldots,X_n\sim\mathrm{Exp}(1)$ being heavily non-normal. The culprit is the Central Limit Theorem.$

Figure 2.1: Histograms of $\sqrt{n}(\bar{X}_n-\mu)$ for independent rv’s $X_1,\ldots,X_n$ with expectation $\mu$ and variance $\sigma^2.$ The pdf of $\mathcal{N}(0,\sigma^2)$ is superimposed in red. A clear convergence appears, despite $X_1,\ldots,X_n\sim\mathrm{Exp}(1)$ being heavily non-normal. The culprit is the Central Limit Theorem.

This chapter introduces the basic elements in statistical inference and proves the simplest case of the central limit theorem.