Page 30 - Probability and Statistical Inference

1. Nations of Probability 7

whole space S belongs to the Borel sigma-field ß since we can write S = ϕ ∈
c
ß, by the requirement (i)-(ii) in the Definition 1.3.2. Also if A ∈ ß for i = 1,2,
i
..., k, then A ∈ ß, since with A = ϕ for i = k+1, k+2, ..., we can express
i i
A as A which belongs to ß in view of (iii) in the Definition 1.3.2.
i i
That is, ß is obviously closed under the operation of finite unions of its mem-
bers. See the Exercise 1.3.1 in this context.
Definition 1.3.3 Suppose that a fixed collection of subsets ß = {A : A ⊆
i
i
S, i ∈ I} is a Borel sigma-field. Then, any subset A of S is called an event if
and only if A ∈ ß.
Frequently, we work with the Borel sigma-field which consists of all sub-
sets of the sample space S but always it may not necessarily be that way. Hav-
ing started with a fixed Borel sigma-field ß of subsets of S, a probability scheme
is simply a way to assign numbers between zero and one to every event while
such assignment of numbers must satisfy some general guidelines. In the next
definition, we provide more specifics.
Definition 1.3.4 A probability scheme assigns a unique number to a set A
∈ ß, denoted by P(A), for every set A ∈ ß in such a way that the following
conditions hold:

Now, we are in a position to claim a few basic results involving probability.
Some are fairly intuitive while others may need more attention.
Theorem 1.3.1 Suppose that A and B are any two events and recall that φ
denotes the empty set. Suppose also that the sequence of events {B ; i = 1}
i
forms a partition of the sample space S. Then,
(i) P(ϕ) = 0 and P(A) = 1;
c
(ii) P(A ) = 1 – P(A);
(iii) P(B ∩ A ) = P(B) – P(B ∩ A);
c
(iv) P(A ∪ B) = P(A) + P(B) – P(A ∩ B);
(v) If A ⊆ B, then P(A) ≤ P(B);
(vi) P(A) = P(A ∩ B ).
i
c
c
Proof (i) Observe that ϕ ∪ ϕ = S and also ϕ, ϕ are disjoint events.
Hence, by part (iii) in the Definition 1.3.4, we have 1 = P(S) = P(ϕ∪ϕ ) =
c
P(ϕ) + P(ϕ ). Thus, P(ϕ) = 1 – P(ϕ = 1 – P(S) = 1 – 1 = 0, in view of part
c
c
(i) in the Definition 1.3.4. The second part follows from part (ii). "