Outcome Variables

Why

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.

Definition

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.

Notation

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 $\set{X \in A, Y \in B}$.

Examples

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 $P[X = a]$. ↩︎