We want to talk about particular attributes of an outcome, instead of the details of the outcomes themselves. These may be useful to specify events.

Given a sample space $\Omega $, an outcome variable (or random variable) is any function on $\Omega $. In this context, the range of the function is called the set of values of the random variable.

Standard convention denotes outcome random variables by capitals $X, Y, Z$ and elements of their codomain by corresponding lower-case, $x, y, z$. Thus, if $X: \Omega \to V$ is a random variable then the lower case $x$ is often reserved for an element of $V$. The event $\Set{\omega \in \Omega }{X(\omega ) = x}$, where $x \in V$, is often abbreviated $\set{X = x}$. The probability of this event is often abbreviated $P(X = x)$.

Similarly, for a subset $A \subset V$, the
event $\Set{\omega \in \Omega }{X(\omega ) \in
A}$ is often abbreviated $\set{X \in A}$ and
its probability abbreviated $P(X \in A)$.^{1}
If $X: \Omega \to V_1$ and $Y: \Omega \to
V_2$, and $A \subset V_1$ and $B \subset V_2$,
then the event $\set{X \in A} \cap \set{Y \in
B}$ is often written $\set{X \in A, Y \in B}$.

*Sum of two dice.*
Suppose we model rolling two dice.
We are interested in the sum of the pips
shown facing up.
Suppose we take as the set of outcome $\set{1,
\dots , 12}$, whose elements correspond to the
sum.
We interpret $\Set{x \in \Omega }{x \geq 10}$
as the event that the sum of the two dice is
greater than or equal to 10.

Alternatively, we may take the usual set of outcomes $\set{1, \dots , 6}^2$ and define an outcome variable $s: \set{1, \dots , 6}^2 \to \set{1, \dots , 12}$ by

\[ s(d_1, d_2) = d_1 + d_2. \]

We interpret this natural-number-valued outcome variable $s$ as sum of the two dice. The event that the sum of the two dice is greater than or equal to to 10 corresponds to the set $\Set{(d_1, d_2) \in \set{1, \dots , 6}}{s(d_1, d_2) \geq 10}$.As a third alternative, again take the usual set of outcomes $\Omega = \set{1, \dots , 6}^2$. Define the outcome variable $X: \Omega \to \N $ to be the number of pips showing on the first die, $Y: \Omega \to \N $ to be the number of pips showing on the second die, and define $Z: \Omega \to \N $ by

\[ Z(\omega ) = X(\omega ) + Y(\omega ) \quad \text{for all } \omega \in \Omega \]

A standard notation for this relation between $Z$ and $X,Y$ by $Z = X + Y$. For example, if $\omega = (2,5)$, $X(\omega ) = 2$, $Y(\omega ) = 5$, and $Z(\omega ) = 7$. The event $\set{Z = 4}$ is the set $\set{(2,2), (1,3), (3,1)}$ If we take the usual distribution $p$ on $\Omega $ with $p(\omega ) = 1/36$ for every $\omega $, the the probability of this event is\[ P(Z = 4) = p(2,2) + p(1,3) + p(3, 1) = 1/36+1/36+1/36 = 1/12 \]

The preceding three paragraphs highlight that there are several ways of denoting the same situation.

*Tossing a fair coin $n$ times.*
As before, we model $n$ tosses of a fair coin
with the sample space $\Omega = \set{0,1}^n$.
Define $X_i: \Omega \to \set{0,1}$ by
$X_i(\omega ) = \omega _i$.
We interpret $\Set{\omega \in
\Omega }{X_i(\omega ) = 1}$ as the event that
toss $i$ turns up heads, and likewise
$\Set{\omega \in \Omega }{X_i(\omega ) = 0}$ as
the event that toss $i$ turns up tails, for $i
= 1, \dots , n$.
We can define the function $X: \Omega \to
{0,1}^n$ by $X(\omega ) = \omega $ or by saying
$X = (X_1, \dots , X_n)$.
Suppose we define $H: \Omega \to \N $ by

\[ H(\omega ) = \sum_{i= 1}^{n} X_i(\omega ) \]

Or, alternatively, $H = \sum_{i = 1}^{n} X_i$. In this case, we interpret $H$ as the number of heads observed in the $n$ tosses.- Occasionally, in the present edition of these sheets, we use the notation $P[X = a]$. ↩︎