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Unformatted text preview: Probability and Statistics Review Introduction to Econometrics Econ 281 Spring Quarter 2007 So what is this all about??? • Let’s assume we want to investigate the following relationships – What is the relationship between student performance and class size? – What is the relationship between education and wages? – Do I earn more if I drink a lot? • No really ! 600 620 640 660 680 700 15 20 25 Test Scores and StudentTeacher Ratio StudentTeacher Ratio Test Scores 10 20 30 40 50 5 10 16 20 Highest Grade Attended Hourly Earning and Education Earnings per hour 12 Population vs. Sample Population (random) Samples Outcomes, Sample Space, and Events • Let’s assume we are dealing with a random process – Number of times your computer crashes in a month – Roll of a die • Outcome = every mutually exclusive result of a random process is an outcome of that process. – Computer crashes 1 time per month – You roll a 6 Outcomes, Sample Space, and Events • Sample Space = the set of all possible outcomes of a random process – Computer can crash 0, 1, 2, 3, 4, 5, 6, ….. times per month – You can roll 1, 2, 3, 4, 5, 6 with a die – Wages: 0 to infinity Outcomes, Sample Space, and Events • Event = subset of the sample space consisting of one of more outcomes – Event = Computer crashes no more than 4 times a month – You roll a 1 or a 4 – NOTE: All outcomes are events but not all events are outcomes S Events and Outcomes = OUTCOMES = EVENTS Random Variables • Random Variable = numerical summary of a random process – Number of times your computer crashes – Number of times you roll a 6 in 10 tries 2 kinds of Random Variables (RVs) • Discrete RVs = has a finite number of possible values = can only take on (discrete) set of values • Continuous RVs = can take on a continuum of possible values Discrete Random Variables • Binary – Gender (male, female) – College Graduate (yes=1/no=0) – Usually coded 0/1 • Multiple Values – Marital Status (Multiple Values: Single, Married, Divorced, Separated) Continuous Random Variables • Examples – Height – Weight – Wages – 10k time – Education Probability Distribution • A Probability Distribution assigns probabilities to events • Example: Number of Heads in 3 coin tosses – Discrete Random variable • Let’s see a graph = = = = = otherwise 3 X if 2 X if 1 X if X if ) ( 8 1 8 3 8 3 8 1 x f Probability Distribution 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 1 2 3 Probability Distribution • For a continuous random variable there are an infinite amount of possible events – Every particular event has probability zero – We can calculate the probability of a range of events • Example: Unemployment duration – P(UED<4weeks) – P(2 weeks < UED < 4 weeks) Cumulative Probability Distribution • The cumulative probability distribution gives the probability that a random variable is less than or equal to a particular value • Example:...
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 Spring '08
 HABERMALZ
 Econometrics, Normal Distribution, Probability distribution, Probability theory, Yi, µx

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