2 Intro: distributions and quantiles

In Science, probability distributions are one of the main ways through which uncertainty about phenomena is modeled and quantified. In the most basic setting, we have a random variable, say \(Y,\) that represents the numerical outcome of an experiment. Formally, \(Y\) is modeled as a measurable function defined on some (abstract/mathematical) measurable space \((\Omega,\mathscr{F}).\) Each probability measure \(\mathbf{P}\) on \((\Omega,\mathscr{F})\) then induces a probability measure on \(\mathbb{R},\) called the distribution of \(Y\) and denoted by \(\mathbf{P}_Y,\) defined through \[\begin{equation} \mathbf{P}_Y(B) := \mathbf{P}[Y\in B] \tag{2.1} \end{equation}\] for each Borel subset \(B\subseteq\mathbb{R}.\) Importantly, as a consequence of Carathéodory’s Extension Theorem, the measure \(\mathbf{P}_Y\) is recoverable from a much simpler function, namely the cumulative distribution function of \(Y,\) denoted by \(F_Y\) (of course, \(F_Y\) depends implicitly on \(\mathbf{P}\)) and defined by \[\begin{equation} F_Y(y) := \mathbf{P}_Y(-\infty,y] = \mathbf{P}[Y\le y],\qquad y\in\mathbb{R}. \end{equation}\] The preceding assertion means that the task of cooking up a probability distribution for a scalar random variable \(Y\) boils down to exhibiting its cumulative distribution function. This is nice because probability measures are not computationally tractable, whereas cumulative distribution functions, being representable through algebraic expressions or at least via numerical formulas, are. To sum up, the message is that if we want to build a model to quantify uncertainty about a scalar phenomenon, all we have to do is to propose a non-decrasing, right-continuous function \(F\colon\mathbb{R}\to\mathbb{R}\) that satisfies the requirements \(\lim_{y\to -\infty}F(y) = 0\) and \(\lim_{y\to+\infty}F(y) = 1.\) Such functions exhaust, in a one-to-one fashion, the category of univariate probability measures.

Example 2.1 Let \(f\colon \mathbb{R}\to \mathbb{R}\) be defined through \[\begin{equation} f(y) := \frac{1}{\sqrt{2\pi}}\mathrm{e}^{-y^2/2},\qquad y\in\mathbb{R}. \end{equation}\] Put \(\Omega = \mathbb{R}\) and let \(\mathscr{F}\) be the class of Borel subsets of \(\Omega.\) Also, define \[\begin{equation} F(y) := \int_{-\infty}^{y} f(u)\,\mathrm{d}u,\qquad y\in \mathbb{R}. \end{equation}\] If we now let \(\mathbf{P}\) be the unique probability measure (given by Carathéodory’s Theorem) on \((\Omega,\mathscr{F})\) satisfying the equality \(\mathbf{P}(-\infty,y] = F(y)\) for all \(y\in\mathbb{R}\) , then \(\mathbf{P}\) is called the standard Gaussian distribution on \(\mathbb{R}\). Additionally, defining the random variable \(Y\) through \[\begin{equation} Y(\omega) := \omega,\qquad \omega\in\mathbb{R}, \end{equation}\] it is clear that \(Y\) has distribution function \(\mathbf{P}\) and cumulative distribution function \(F,\) that is, \(\mathbf{P}_Y = \mathbf{P}\) and \(F_Y = F.\) Such \(Y\) is called a standard Gaussian (or: standard Normal) random variable.