Introduction to the Mathematical and Statistical Foundations of Econometrics
Probability Measure
Let us return to the Texas lotto example. The odds, or probability, of winning are 1 /N for each valid combination rnj of six numbers; hence, if you play n different valid number combinations {«jl, ...,Mjn}, the probability of winning is n/N:P({«^,}) = n/N. Thus, in the Texas lotto case the probability P(A), A e &, is given by the number n of elements in the set A divided by the total number N of elements in ^. In particular we have P(Q) = 1, and if you do not play at all the probability of winning is zero: P(0) = 0.
The function P(A), A e &, is called a probability measure. It assigns a number P(A) e [0, 1] to each set A e &. Not every function that assigns numbers in [0,1] to the sets in & is a probability measure except as set forth in the following definition:
Definition 1.3: A mapping P: & ^ [0, 1] from a a-algebra & of subsets of a set ft into the unit interval is a probability measure on {ft, &} if it satisfies the following three conditions:
For all A e &, P(A) > 0,
P (ft) = 1,
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Д A0=£P (-A)
Recall that sets are disjoint if they have no elements in common: their intersections are the empty set.
The conditions (1.8) and (1.9) are clearly satisfied for the case of the Texas lotto. On the other hand, in the case under review the collection & of events contains only a finite number of sets, and thus any countably infinite sequence of sets in & must contain sets that are the same. At first sight this seems to conflict with the implicit assumption that countably infinite sequences of disjoint sets always exist for which (1.10) holds. It is true indeed that any countably infinite sequence of disjoint sets in a finite collection & of sets can only contain a finite number of nonempty sets. This is no problem, though, because all the other sets are then equal to the empty set 0. The empty set is disjoint with itself, 0П0 = 0, and with any other set, A n0 = 0. Therefore, if & is finite, then any countable infinite sequence of disjoint sets consists of a finite number of nonempty sets and an infinite number of replications of the empty set. Consequently, if & is finite, then it is sufficient to verify condition (1.10) for any pair of disjoint sets Ai, A2 in &, P(Ai U A2) = P(Ai) + P(A2). Because, in the Texas lotto case P(A1 U A2) = (n1 + n2)/N, P(A1) = n1/N, and P(A2) = n2/N, where n1 is the number of elements of A1 and n2 is the number of elements of A2, the latter condition is satisfied and so is condition (1.10).
The statistical experiment is now completely described by the triple {ft, &, P}, called the probability space, consisting of the sample space ft (i. e., the set of all possible outcomes of the statistical experiment involved), a a - algebra & of events (i. e., a collection of subsets of the sample space ft such that the conditions (1.5) and (1.7) are satisfied), and a probability measure P: & ^ [0, 1] satisfying the conditions (1.8)-(1.10).
In the Texas lotto case the collection & of events is an algebra, but because & is finite it is automatically a a-algebra.
