Real-Valued Outcome Variables

Why

The set of real numbers is large, and so we can often embed other sets within it. Also, we are often interested in modelling random quantities.

Definition

A real outcome variable (or real random variable, random variable) is an outcome variable whose codomain is a subset of the real numbers. Such variables are often called quantitative. Caution: many authorities reserve the term random variable for outcome variables whose domain is $\R $.

The probability mass function (or p.m.f., pmf) of a random variable $X: \Omega \to \R $ is the function $f: \R \to \R $ defined by

\[ f(x) = P(X = x) \]

If $\Omega $ is finite, then $\range X$ is a finite set, and so the probability mass function is the extension to $\R $ of the induced distribution $p: \range(X) \to \R $ of $X$. If $\range(X)$ is finite or countable, we call $X$ a discrete random variable.

Notation

For a real-valued random variable $X: \Omega \to \R $ and $\alpha \in \R $, we often abbreviate the sets

\[ \Set{\omega \in \Omega }{X(\omega ) \leq \alpha } \text{ and } \Set{\omega \in \Omega }{X(\omega ) \geq \alpha } \]

by $\set{X \leq \alpha }$ and $\set{X \geq \alpha }$ respectively. Also, given a probability measure $P$, we denote the probabilities of these events by $P(X \leq \alpha )$ and $P(X \geq \alpha )$, respectively. Similar to before, the notation $X \sim f$ is shorthand for the random variable $X: \Omega \to \R $ has probability mass function $f: \R \to \R $.

Examples

Tossing a fair coin $n$ times. Suppose we model $n$ tosses of a fair coin as usual, so that $\Omega = \set{0,1}^n$ and $p: \Omega \to \R $ is defined by

\[ p(\omega ) = 2^{-n} \quad \text{for all } \omega \in \Omega \]

Recall that we have calculuated $P(X = k)$ to be ${n \choose k} 2^{-n}$. Thus, the probability mass function $f: \R \to \R $ of $X$ satisifes

\[ f(k) = {n \choose k} 2^{-n} \quad \text{for } k = 0, \dots , n \]

and $f(x) = 0$ for $x \neq 0, \dots , n$.

Cumulative distribution function

Given a random variable $X: \Omega \to \R $ and probability measure $P$ on $\pow{\Omega }$, the function $F: \R \to \R $ defined by

\[ F(x) = P(X \leq x) \]

is called the cumulative distribution function of $X$.