eco331-01-Intro - Northwestern University Marciano...

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Unformatted text preview: Northwestern University Marciano Siniscalchi Fall 2009 Econ 331-0 INTRODUCTION AND REVIEW OF PROBABILITY 1. Introduction This course focuses on economic decisions in circumstances where the ultimate consequences of an individual’s actions are cannot be known in advance. More succinctly, we say that such decisions are made “under risk” or “choice uncertainty” (there is a distinction, but we won’t worry about it just yet). Our emphasis will be on decisions under uncertainty, not uncertainty in and of itself (statistics, on the other hand, focuses on uncertainty). But, clearly, we need to develop a language that allows us to talk about uncertainty in precise terms. This is the language of probability theory , which we shall review now. 2. Probability Theory: A Review First, I need to remind you of a few definitions from set theory . You may wish to keep these notes handy and refer to them from time to time. In case you were wondering, the reason we care about sets so much is that they provide an extremely convenient, simple representation of uncertain “facts”. Note for the impatient: if you already know some set theory, you may wish to skip directly to Subsection 2.2, and refer to Subsection 2.1 only as needed. 2.1. Definitions from Set Theory. A set is a collection of objects, called elements of the set. One defines a set either writing down its elements one by one, or by stating properties that make them legitimate members of the set in question. For instance, X = { 1 , 2 , 3 , 4 , 5 , 6 } and Y = { n : n is a natural number smaller than 7 } are valid definitions of sets, and of course X = Y . Also, if a set contains infinitely many elements, one clearly cannot write out all its elements explicitly; so, to define infinite sets, one must necessarily adopt the second approach, i.e. state the defining properties of its elements. Example: E = { n : n is an even natural number } . The empty set ∅ is a special set that has no element. In order to indicate that the object e is an element of the set E , I write e ∈ E . In order to indicate that e is not an element of E , I write e 6∈ E . A set E is a subset of another set F , written E ⊆ F , if and only if every element e of E is also an element of F . More concisely: E ⊆ F if and only if e ∈ E implies e ∈ F . Clearly, if E ⊆ F and F ⊆ E , then E = F . The notation E ⊂ F indicates that E ⊆ F , but it is not the case that F ⊆ E ; then, E is said to be a strict subset of F . Elements of sets can be anything; for instance, they can themselves be sets! In particular, for any set X , I will denote by P ( X ) the set of all subsets of X —the so-called power set of X . Example: if X = { 1 , 2 , 3 } , then P ( X ) = {∅ , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 }} ....
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This note was uploaded on 11/17/2009 for the course ECONOMICS 331 taught by Professor Marciano during the Spring '09 term at Northwestern.

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eco331-01-Intro - Northwestern University Marciano...

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