# Chapter 2 Density estimation

A random variable \(X\) is completely characterized by its cdf. Hence, an estimation of the cdf yields as a side-product estimates for different characteristics of \(X\) by plugging-in \(F_n\) in the \(F\). For example, the mean \(\mu=\mathbb{E}[X]=\int x \,\mathrm{d}F(x)\) can be estimated by \(\int x \,\mathrm{d}F_n(x)=\frac{1}{n}\sum_{i=1}^n X_i=\bar X\). Despite their usefulness, cdfs are hard to visualize and interpret.

**Densities**, on the other hand, are easy to visualize and interpret, making them **ideal tools for data exploration**. They provide immediate graphical information about the most likely areas, modes, and spread of \(X\). A *continuous* random variable is also completely characterized by its pdf \(f=F'\). Density estimation does not follow trivially from the ecdf \(F_n\), since this is not differentiable (not even continuous), hence the need of the specific procedures we will see in this chapter.