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Probability Theory: Homework 1

Exercises, due Wed Sept 3.

From Jacod and Protter, problems 2.6, 2.7, 2.10 and 2.14.

2.6 Let \mathcal{A} be a \sigma-algebra of subsets of \Omega and let B \in \mathcal{A}. Show that \mathcal{F} := \{A \cap B \, : \, A \in \mathcal{A}\} is a \sigma-algebra of subsets of B.  Is this still true if B is a subset of \Omega that is not a member of \mathcal{A}?

2.7  Let f be a function mapping \Omega to another space E with a \sigma-algebra \mathcal{E}. Let \mathcal{A} := \{A \subset \Omega \, : \, \text{there exists } B \in \mathcal{E} \text{ with } A = f^{-1}(B)\}. Show that \mathcal{A} is a \sigma-algebra on \Omega.

2.10 Using the definition of a probability measure, show that if A, \, B\, \in \mathcal{A}, then \mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B) - \mathbb{P}(A \cap B).

2.14 Suppose that \mathbb{P}(A) = \frac{3}{4} and \mathbb{P}(B) = \frac{1}{3}. Show that \frac{1}{12} \leq \mathbb{P}(A \cap B) \leq \frac{1}{3}.

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