Random Variable

Definition 11.1 (Random variable). A random variable $X$ is a function defined on a sample space $S$ into the real numbers $\mathbb R$, such that the inverse image of every point, subset or interval of $\mathbb R$ is an event in $S$ — i.e. a set to which the underlying experiment already assigns a probability.

Notation: capital letters $X,Y,Z,\dots$ denote random variables (the function itself); the corresponding small letters $x,y,z,\dots$ denote the possible values the random variable can take. Formally $X:S\to\mathbb R$, and for a sample point $\omega\in S$, $X(\omega)$ is the real number assigned to it. The range set (or support) of $X$ is $R_X=\{X(\omega):\omega\in S\}$ — the collection of values $X$ actually takes.

Inverse image. If $x$ is a value of $X$, its inverse image $X^{-1}(x)=\{\omega\in S: X(\omega)=x\}$ is the set of all sample points mapping to $x$ — and this set is always an event of $S$, so it has a well-defined probability. This is the mechanism that lets us talk about "the probability that $X$ equals $x$": it is really the probability of the underlying event $X^{-1}(x)$.

Worked illustrations.

• Coin toss. $S=\{H,T\}$. Let $X$ = number of heads: $X(T)=0,\ X(H)=1$. So $X$ is a random variable taking the values $0$ and $1$, and e.g. $X^{-1}(1)=\{H\}$.
• Two coins (finding an inverse image). $S=\{HH,HT,TH,TT\}$, $X=$ number of tails. Then $X(TT)=2,\ X(HT)=X(TH)=1,\ X(HH)=0$; $X$ takes the values $0,1,2$ with $1,2,1$ sample points in their inverse images respectively (total $4$, matching $|S|$).
• Two dice. $S$ has $36$ ordered pairs $(\alpha,\beta)$, $1\le\alpha,\beta\le6$; $X=\alpha+\beta$ takes the values $2,3,\dots,12$, with the familiar triangular counts $1,2,3,4,5,6,5,4,3,2,1$ (summing to $36$). …