Sample Spaces and Events PROBABILITY

40 Sample Spaces and Events PROBABILITY

Let S be a sample space which consists of the possible outcomes of an experiment where the events are subsets of S. The sample space S itself is called the certain event, and the null set ∅ is called the impossible event .

It would be convenient if all subsets of S could be events. Unfortunately, this may lead to contradictions when a probability function is defined on the events. Thus, the events are defined to be a limited collection

C of subsets of S as follows. DEFINITION 40.1: The class C of events of a sample space S form a σ-field. That is, C has the following three

properties: (i) S ∈ C.

(ii) If A 1 ,A 2 , … belong to C, then their union A 1 ∪A 2 ∪A 3 ∪ … belongs to C. (iii) If A ∈ C, then its complement A c ∈ C.

Although the above definition does not mention intersections, DeMorgan’s law (40.3) tells us that the complement of a union is the intersection of the complements. Thus, the events form a collection that is closed under unions, intersections, and complements of denumerable sequences.

If S is finite, then the class of all subsets of S form a σ-field. However, if S is nondenumerable, then only certain subsets of S can be the events. In fact, if B is the collection of all open intervals on the real line R, then the smallest σ-field containing B is the collection of Borel sets in R.

If Condition (ii) in Definition 40.1 of a σ-field is replaced by finite unions, then the class of subsets of S is called a field. Thus a σ-field is a field, but not visa versa. First, for completeness, we list basic properties of the set operations of union, intersection, and complement.

40.1. Sets satisfy the properties in Table 40-1.

TABLE 40-1

Laws of the Algebra of Sets

Idempotent laws: (1a) A ∪A=A

(1b) A ∩A=A

Associative laws: (2a) (A ∪ B) ∪ C = A ∪ (B ∪ C) (2b) (A ∩ B) ∩ C = A ∩ (B ∩ C) Commutative laws: (3a) A ∪ B = B ∪ A (3b) A ∩B=B∩A Distributive laws: (4a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (4b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ B) Identity laws:

(5a) A ∪∅=A

(5b) A ∩U=A

(6b) A ∩∅=∅ Involution law:

(6a) A ∪U=U

(7) (A C ) C =A

Complement laws: (8a) A

c ∪A c =U (8b) A ∩A =∅

(9a) U c =∅ c (9b) ∅ =U

DeMorgan’s laws: (10a) (A ∪ B) c =A c ∩ B c (10b) (A ∩ B) c =A c ∪ B c

PROBABILITY

40.2. The following are equivalent: (i) A ⊆ B, (ii) A ∩ B = A, (iii) A ∩ B = B. Recall that the union and intersection of any collection of sets is defined as follows:

∪ j A j = {x | there exists j such that x ∈ A j } and ∩ j A j = {x | for every j we have x ∈ A j }

40.3. (Generalized DeMorgan’s Law) (10a)'(∪ A ) j c

j A j c ; (10b)'(∩ j A j ) j c =∩ =∪ A j c j

Probability Spaces and Probability Functions DEFINITION 40.2: Let P be a real-valued function defined on the class C of events of a sample space S. Then P is

called a probability function, and P(A) is called the probability of an event A, when the following axioms hold: Axiom [P 1 ] For every event A, P(A) ≥ 0.

Axiom [P 2 ] For the certain event S, P(S) = 1.

Axiom [P 3 ] For any sequence of mutually exclusive (disjoint) events A 1 ,A 2 , …, P(A 1 ∪A 2 ∪ …) = P(A 1 ) + P(A 2 ) +…

The triple (S, C, P), or simply S when C and P are understood, is called a probability space. Axiom [P 3 ] implies an analogous axiom for any finite number of sets. That is:

Axiom [P 3 '] For any finite collection of mutually exclusive events A 1 ,A 2 , …, A n , P(A 1 ∪ A 2 ∪ … ∪ A n ) = P(A 1 ) + P(A 2 ) + … + P(A n )

In particular, for two disjoint events A and B, we have P(A ∪ B) = P(A) + P(B).

The following properties follow directly from the above axioms.

40.4. (Complement rule) P(A c ) = 1 – P(A). Thus, P( ∅ ) = 0.

40.5. (Difference Rule) P(A\B) = P(A) – P(A ∩ B).

40.6. (Addition Rule) P(A ∪ B) = P(A) + P(B) – P(A ∩ B).

40.7. n For n ≥ 2, P ∪

j =1 A j ≤ ∑ j = 1 PA ( () ) j

40.8. (Monoticity Rule) If A ⊆

B, then P(A) ≤ P(B).

Limits of Sequences of Events

40.9. (Continuity) Suppose A 1 ,A 2 , … form a monotonic increasing (decreasing) sequence of events; that is, j A ⊆A j +1 (A j ⊇A j +1 ). Let A =∪ j A j (A =∩ j A j ). Then lim P(A n ) exists and

lim P(A n ) = P(A)

For any sequence of events A 1 ,A 2 , …, we define

n = ∪ k = 1 ∩ jk = A j and lim sup A n = ∩ k = 1 ∪ jk = A j

If lim inf A n = lim sup A n , then we call this set lim A n . Note lim A n exists when the sequence is monotonic.

PROBABILITY

40.10. For any sequence A j of events in a probability space,

P(lim inf A n ) ≤ lim inf P(A n ) ≤ lim sup P(A n ) ≤ P(lim sup A n )

Thus, if lim A n exists, then P(lim A n ) = lim P(A n ).