Section 1.4 - The Single-Element Model

A trial X produces an outcome drawn from a finite set of possible outcomes { x_i }, 0 ≤ i ≤ n.^?For example, if X represents the roll of a single die, the set of possible outcomes is { x_i } for 1 ≤ i ≤ 6, where x_i represents the outcome “the upper-most face of the die shows exactly i spots”. The set of possible outcomes of X, { x_i }, is known as the sample space, sometimes also called X. A single trial outcome can be generalized to a sequence containing finitely, countably, or non-countably many outcomes representing a similarly-sized sequence of trials.

Given a trial X, the first task is to find the sample space. Do not confuse the map and the territory: abstraction is at play when defining the sample space for X.^?For instance, to continue the die example, a tossed die may get caught in a hole and land point up. However, such an occurrence is so unlikely that most die-roll models can ignore it. In addition, x_i is not a number; the operation x_i + x_j doesn’t make any sense if + is the usual integer addition.