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notes1 - Stat 704 Data Analysis I Probability Review...

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Stat 704 Data Analysis I Probability Review Timothy Hanson Department of Statistics, University of South Carolina 1 / 29
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Course information Logistics: LeConte College 210A, Tuesday & Thursday 3:30-4:45pm. Instructor: Tim Hanson, Leconte 219C, phone 777-3859. Office hours: Tuesday/Thursday 9-10:30am and by appointment. Required text: Applied Linear Statistical Models (5th Edition), by Kutner, Nachtsheim, Neter, and Li. Online notes at http://www.stat.sc.edu/ hansont/stat704/stat704.html Grading: homework 50%, two exams 25% each. Stat 704 has a co-requisite of Stat 712 (Casella & Berger level mathematical statistics). You need to be taking this, or have taken this already. 2 / 29
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A.3 Random Variables def’n : A random variable is defined as a function that maps an outcome from some random phenomenon to a real number. More formally, a random variable is a map or function from the sample space of an experiment, S , to some subset of the real numbers R R . Restated: A random variable measures the result of a random phenomenon. Example 1 : The height Y of a randomly selected University of South Carolina statistics graduate student. Example 2 : The number of car accidents Y in a month at the intersection of Assembly and Gervais. 3 / 29
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cdf, pdf, pmf Every random variable has a cumulative distribution function (cdf) associated with it: F ( y ) = P ( Y y ) . Discrete random variables have a probability mass function (pmf) f ( y ) = P ( Y = y ) = F ( y ) - F ( y - ) = F ( y ) - lim x y - F ( x ) . (A.11) Continuous random variables have a probability density function (pdf) such that for a < b P ( a Y b ) = integraldisplay b a f ( y ) dy . For continuous random variables, f ( y ) = F ( y ) . Question : Are the two examples on the previous slide continuous or discrete? 4 / 29
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A.3 Expected value The expected value , or mean of a random variable is, in general, defined as E ( Y ) = integraldisplay −∞ y dF ( y ) . For discrete random variables this is E ( Y ) = summationdisplay y : f ( y ) > 0 y f ( y ) . (A.12) For continuous random variables this is E ( Y ) = integraldisplay −∞ y f ( y ) dy . (A.14) 5 / 29
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E ( · ) is linear Note : If a and c are constants, E ( a + cY ) = a + cE ( Y ) . (A.13) In particular, E ( a ) = a E ( cY ) = cE ( Y ) E ( Y + a ) = E ( Y ) + a 6 / 29
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A.3 Variance The variance of a random variable measures the “spread” of its probability distribution. It is the expected squared deviation about the mean : var ( Y ) = E { [ Y - E ( Y )] 2 } (A.15) Equivalently, var ( Y ) = E ( Y 2 ) - [ E ( Y )] 2 (A.15a) Note : If a and c are constants, var ( a + cY ) = c 2 var ( Y ) (A.16) In particular, var ( a ) = 0 var ( cY ) = c 2 var ( Y ) var ( Y + a ) = var ( Y ) Note : The standard deviation of Y is sd ( Y ) = radicalbig var ( Y ) . 7 / 29
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Example Suppose Y is the high temperature in Celsius of a September day in Seattle. Say E ( Y ) = 20 and var ( Y ) = 10. Let W be the high temperature in Fahrenheit. Then E ( W ) = E parenleftbigg 9 5 Y + 32 parenrightbigg = 9 5 E ( Y ) + 32 = 9 5 20 + 32 = 68 degrees .
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