# Chap06 - Chapter 6 Fundamentals of Probability and...

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1 Chapter 6 Fundamentals of Probability and Statistics

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2 1. Preliminaries Sample space Events 2. Probability Axioms Axiom 3 3. Conditional Probability [Summary: 4 rules of probability] 1. Independence Random Sample 2. Simpson’s Paradox
3 II. The World: Probability Theory (Ch. 6 – 8) III. Drawing Conclusions: Statistical Inference (Ch. 9 – 11) I. Data: Descriptive Statistics (Ch. 2 – 5) X Y Z 33 2 300 33 2 300 29 1 1538 26 1 1538 32 1 1538 27 1 1538 32 1 420 22 1 460 50 1 1211.53 54 1 1096.15 25 2 288.46 20 2 134.61 39 4 250 37 4 250 58 1 1923.07 61 1 2884

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4 Theoretical Underpinnings So far, we have only explored our data We also want to figure out where this data came from. I.e., we want a mathematical model of that part of “nature” that most likely produced the data.
5 Theoretical Underpinnings “Nature” can be: A coin that will be flipped. the distribution of all (possible) consumers of a certain sort. The distribution of all (possible) revenues for a given day. The distribution of all (possible) number of books in childrens’ houses.

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6 Consider the picture illustrating the relation between “nature” (represented by a probability distribution) and our “evidence” (i.e., a data set) If we know “nature” (the probability distribution), we can figure out the probability of obtaining any given data set. This is “probability theory” If we have a particular data set, we can try to figure out which probability distributions would most likely have produced it. This is inferential statistics (part III, chaps. 9 – 11)
7 1. A sample space is the set of all the possible “elementary” outcomes of a single trial, experiment, etc. (p. 174) E.g., the sample space of one toss of a normal die is: {1, 2, 3, 4, 5, 6} More precisely, a sample space is the set of all outcomes that have positive probability (or density – more on this later). Two Preliminary Definitions

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8 Sample Spaces Drawing a poker card from a deck {A : , 2 : , 3 : ,…, Q , K } How many of 30 children have measles? {0, 1, 2, …, 29, 30} Flipping a (possibly biased) coin twice {HH, HT, TH, TT} Net income [0, ) Net worth (– , )
9 Sample Spaces All the events in a sample space are mutually exclusive : only one of them can occur in any given trial The set of events is exhaustive : in any given trial, one of these events must occur. All the events in the sample space are possible ; none have zero probability. These are the three key features of a sample space Hence, in any given trial, one and only one of the events in the sample space occurs. The sample space does not indicate how probable the various outcomes are.

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10 1. An event is any set of outcomes in the given sample space. Sample space Example events {A : , 2 : , 3 : ,…, Q , K } “Drawing a 7” = {7 : 7 , 7 , 7 } {0, 1, 2, …, 29, 30} “No more than 3” = {0, 1, 2, 3} {HH, HT, TH, TT} “One Heads” = {HT, TH} [0, ) “Actual income” = {87,642.98}
11 Probability Theory To get started, let’s assume we have a finite sample space with n possible outcomes: {e 1 ,…, e n }

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• The probability of any one of these event e i is
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## This note was uploaded on 01/20/2011 for the course ECON 15A taught by Professor Shirey during the Winter '08 term at UC Irvine.

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Chap06 - Chapter 6 Fundamentals of Probability and...

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