Page 75 - Probability and Statistical Inference

52 1. Notions of Probability

1.3.3 Suppose that A , ..., A are disjoint Borel sets. Then show that
1
k
. Contrast this result with the requirement in part (iii)
of the Definition 1.3.4.
1.3.4 Prove part (v) in the Theorem 1.3.1.
1.3.5 Consider a sample space S and suppose that ß is the Borel sigma-
field of subsets of S. Let A, B, C be events, that is these belong to ß. Now,
prove the following statements:
(i) P(A ∪ B) = P(A) + P(B);
(ii) P(A ∩ B) = P(A) + P(B) – 1;
(iii) P(A ∩ C) = min{P(A), P(C)};
(iv) P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B)
– P(A ∩ C) – P(B ∩ C) + P(A ∩ B ∩ C).
{Hint: Use the Theorem 1.3.1 to prove Parts (i)-(iii). Part (iv) should fol-
low from the Theorem 1.3.1, part (iv). The result in part (ii) is commonly
referred to as the Bonferroni inequality. See the Theorem 3.9.10.}
1.3.6 (Exercise 1.3.5 Continued) Suppose that A , ..., A are Borel sets,
that is they belong to ß. Show that 1 n

{Hint: Use mathematical induction and part (iv) in the Exercise 1.3.5.}
1.3.7 (Exercise 1.3.5 Continued) Suppose that A , ..., A are Borel sets, that
n
1
is they belong to ß. Show that

This is commonly referred to as the Bonferroni inequality. {Hint: Use math-
ematical induction and part (ii) in the Exercise 1.3.5. See the Theorem 3.9.10.}
1.3.8 Suppose that A , A are Borel sets, that is they belong to ß. Show that
1 2

where recall from (1.2.1) that A ∆A stands for the symmetric difference of
1
2
A , A .
1 2
1.3.9 Suppose that A , A , A are Borel sets, that is they belong to ß. Recall
3
2
1
from (1.2.1) that A ∆ A stands for the symmetric difference of A , A . Show that
i j i j
(i) A ∆A ⊆ (A ∆A ) ∪ (A ∆A );
1 3 1 2 2 3
(ii) P(A ∆A ) ≤ P(A ∆A ) + P(A ∆A ).
1 3 1 2 2 3