1 Questions for Chapter 1

1.1 Warm-up

Exercise 1.1 Consider the sample space \(\Omega=\{1,2,3,4,5,6,7,8,9\}\). Let \[\begin{align*} A&=\{2,4,6,8\}, & B&=\{3,6,9\}, & C&=\{2,3,5,7\}. \end{align*}\] Write down the corresponding sets for the following events:

\(A\) and \(B\);
\(C\) or \(A\);
\(B\) minus \(C\);
not \(A\).

Exercise 1.2 Prove (C2), using just (A2) and (C1).

Exercise 1.3 Prove (C3), using just (A2) and (C2).

Exercise 1.4 Prove (C4), using just (A1) and (C2).

Exercise 1.5 On each day of a certain week, you are to observe whether it rains. Arrange the following events in order of increasing probability, with an explanation of your answer. Do not make any assumptions of independence. \[\begin{align*} E_1 & \text{ is the event that it rains on Monday.}\\ E_2 & \text{ is the event that it rains on Monday and Tuesday.}\\ E_3 & \text{ is the event that it rains on Monday, Tuesday and Wednesday, but not on Thursday.}\\ E_4 & \text{ is the event that it rains on Monday and Tuesday, but not on Thursday}. \end{align*}\]

Exercise 1.6 Prove (C10), using just (A2) and (C7).

Exercise 1.7 There is a probability of \(3/4\) that Eric goes to the pub on Sunday evening (event \(A\)), and there is a probability of \(1/3\) that Eric arrives late to work on Monday morning (event \(B\)). Without making any assumptions on the dependence of these two events, show that \[ \frac{1}{12} \leq \mathbb{P}(A \cap B) \leq \frac{1}{3}. \]

Exercise 1.8 Anne and Bob meet on ‘Blind Date’. If the probability that Anne falls in love with Bob is 0.4, the probability that Bob falls in love with Anne is 0.2 and the probability that both Anne and Bob fall in love with each other is 0.1, then find the probability that

at least one person falls in love;

hint:use (C6)
neither person falls in love;

hint:use (C2) plus part a.
exactly one of the two people fall in love.

hint:use (C1) plus part a.

Exercise 1.9 Consider the sample space \(\Omega = \{1,2,3,4,5,6\}\). Which of the following are \(\sigma\)-algebras, and which are not? Justify your answers.

\(\mathcal{F}_1 = \{ \emptyset, \{1,2,3\} , \{4,5,6\}, \Omega \}\);
\(\mathcal{F}_2 = \{ \emptyset, \{ 1,2,3 \}, \{4, 5\}, \{ 6\}, \Omega \}\);
\(\mathcal{F}_3 = \{ \emptyset, \{1, 2, 3, 4 \}, \{1,2,3,4,5\}, \{1,2,3,4,6\}, \{ 5 \} , \{ 6 \}, \{5,6\}, \Omega \}\).

1.2 Workout

Exercise 1.10 Prove (C1), using just (A3).

Exercise 1.11 Prove (C5), using just (A1) and (C1).

Exercise 1.12 Prove (C6), using just (A3) and (C1).

Exercise 1.13 Using (C2) and (C8), prove Boole’s other inequality: for events \(A_1,A_2,\ldots\), \[ \mathbb{P}\left( \bigcap_{i=1}^\infty A_i \right) \geq 1 - \sum_{i=1}^\infty \mathbb{P}( A_i^{\mathrm{c}}) .\]

Exercise 1.14 A special case of (C8) (Boole’s inequality) is that for any finite collections of sets \(A_1, A_2, \ldots, A_k\), \[ \mathbb{P}( \bigcup_{i=1}^k A_i ) \leq \sum_{i=1}^k \mathbb{P}(A_i) .\] Prove this result directly, using just (A1) and (C6), by induction.

Exercise 1.15 Using (C6), prove that for events \(A, B\), and \(C\) we have \[ \mathbb{P}(A \cup B \cup C) = \mathbb{P}(A) + \mathbb{P}(B) + \mathbb{P}(C) - \mathbb{P}(A \cap B) - \mathbb{P}(B \cap C) - \mathbb{P}(A \cap C) + \mathbb{P}(A \cap B \cap C) .\]

Exercise 1.16 Suppose that \(\Omega\) is finite and \(A_1, \ldots, A_n\) is a partition of \(\Omega\) (so the \(A_i\) are non-empty, pairwise disjoint, and \(\bigcup_i A_i = \Omega\)). Let \(\mathcal{F}\) be the set of all possible unions of some of the \(A_i\)s (i.e., the sets \(\bigcup_{i \in I} A_i\) for all \(I \subseteq \{1,2,\ldots,n\}\), a total of \(2^n\) sets in all). Verify that \(\mathcal{F}\) is a \(\sigma\)-algebra.

Exercise 1.17 Let \(\mathcal{F}\) and \(\mathcal{G}\) be two \(\sigma\)-algebras over \(\Omega\). Show that their intersection \(\mathcal{F} \cap \mathcal{G}\) is also a \(\sigma\)-algebra over \(\Omega\).

Exercise 1.18 Show that a \(\sigma\)-algebra \(\mathcal{F}\) is closed under countable intersections: i.e., if \(A_1, A_2, \ldots \in \mathcal{F}\), then \(\bigcap_{n=1}^\infty A_n \in \mathcal{F}\).

Let \(\mathcal{F}\) be a \(\sigma\)-algebra over \(\Omega\) and suppose that \(B \in \mathcal{F}\). Consider \(\mathcal{G} = \{ A \cap B : A \in \mathcal{F} \}\), and show that \(\mathcal{G}\) is a \(\sigma\)-algebra over \(B\).

1.3 Stretch

Exercise 1.19 Consider the scenario of tossing a coin repeatedly and indefinitely. Consider the sample space \(\Omega\) being the set of all sequences \(\omega = (\omega_1, \omega_2, \ldots )\) where \(\omega_i\) is the outcome of the \(i\)th roll, H or T. The notation for this is \(\Omega = \{\text{H},\text{T}\}^{\mathbb{N}}\). Consider the events \(E_i = \{ \omega_i = \text{H} \}\) and let \[ A = \{ \text{roll H infinitely often} \} = \cap_{n=1}^\infty \cup_{m=n}^\infty E_m .\] Express the event \(A^{\mathrm{c}}\) (i) in symbols, and (ii) in words.

Exercise 1.20 Prove (C7), using just (A3), by induction.

Exercise 1.21 Prove (C8), using just (A4) and (C5).

Exercise 1.22 The purpose of this exercise is to prove that (A4) (countable additivity) is equivalent to (C7) (finite additivity) plus (C9) (continuity along monotone limits).

Show that (A4) implies (C9).

hint: first show that it suffices to establish the first half of C9 (for increasing events); then express \(\cup_{m=1}^\infty A_m\) as a countable union of pairwise disjoint events.
Show that (C7) and (C9) together imply (A4).

Exercise 1.23 Suppose \(A\), \(B\), \(C\) are events with \[\begin{align*} \mathbb{P}(A) &= 0.5, & \mathbb{P}(B)&= 0.7, & \mathbb{P}(C) &= 0.6, \\ \mathbb{P}(A\cap B) &= 0.3, & \mathbb{P}(B\cap C)&= 0.4, & \mathbb{P}(C \cap A) &= 0.2, \\ \mathbb{P}( A \cap B \cap C) &= 0.1. \end{align*}\]

Find the probability that exactly two of \(A\), \(B\), \(C\) occur.
Find the probability that exactly one of \(A\), \(B\), \(C\) occurs.

Exercise 1.24 Let \(\mathcal{F}_1 \subseteq \mathcal{F}_2 \subseteq \cdots\) be an increasing sequence of \(\sigma\)-algebras over a sample space \(\Omega\). Consider \(\mathcal{F} = \bigcup_i \mathcal{F}_i\).

Show that \(\mathcal{F}\) satisfies (S1) and (S2), and that for any \(A, B \in \mathcal{F}\), we have \(A \cup B \in \mathcal{F}\) (this is weaker than (S3).
(Hard!) Show that \(\mathcal{F}\) need not be a \(\sigma\)-algebra by considering the example where \(\Omega = \mathbb{N}\) and where \(\mathcal{F}_n\) consists of all subsets of \(\{1,2,\ldots,n\}\) and their complements in \(\mathbb{N}\).