Introduction to the Mathematical and Statistical Foundations of Econometrics
Sample Space
The Texas lotto is an example of a statistical experiment. The set of possible outcomes of this statistical experiment is called the sample space and is usually denoted by ^. In the Texas lotto case, ^ contains N = 15,890,700 elements: ^ = {&>imn}, where each element Mj is a set itself consisting of six different numbers ranging from 1 to 50 such that for any pair юі, Mj with i = j, юі = юj. Because in this case the elements Mj of ^ are sets themselves, the condition юі = юj for i = j is equivalent to the condition that юі П Mj ^ ^.
1.1.2. Algebras and Sigma-Algebras of Events
A set {Mj1,Mjt} of different number combinations you can bet on is called an event. The collection of all these events, denoted by Ж, is a “family” of subsets of the sample space ^. In the Texas lotto case the collection Ж consists of all subsets of Й, including ^ itself and the empty set 0.4 In principle, you could bet on all number combinations if you were rich enough (it would cost you $15,890,700). Therefore, the sample space ^ itself is included in Ж. You could also decide not to play at all. This event can be identified as the empty set 0. For the sake of completeness, it is included in Ж as well.
Because, in the Texas lotto case, the collection & contains all subsets of Й, it automatically satisfies the conditions
If A e & then A = tiA e &, (1.5)
where A = ^A is the complement of the set A (relative to the set Й), that is, the set of all elements of ^ that are not contained in A, and
If A, B e & then A U B e &. (1.6)
By induction, the latter condition extends to any finite union of sets in &: If Aj e & for j = 1,2,..., n, then Un=j Aj e &.
Definition 1.1: A collection & of subsets of a nonempty set ^ satisfying the conditions (1.5) and (1.6) is called an algebra.[5]
In the Texas lotto example, the sample space ^ is finite, and therefore the collection & of subsets of ^ is finite as well. Consequently, in this case the condition (1.6) extends to
TO
If Aj e & for j = 1,2, 3,... then U Aj e &. (1.7)
j=1
However, because in this case the collection & of subsets of ^ is finite, there are only a finite number of distinct sets Aj e &. Therefore, in the Texas lotto case the countable infinite union U°=j Aj in (1.7) involves only a finite number of distinct sets Aj; the other sets are replications of these distinct sets. Thus, condition (1.7) does not require that all the sets Aj e & are different.
Definition 1.2: A collection & of subsets of a nonempty set ^ satisfying the conditions (1.5) and (1.7) is called a a-algebra.[6]
