MIT14_30s09_lec16

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MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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14.30 Introduction to Statistical Methods in Economics Lecture Notes 16 Konrad Menzel April 9, 2009 1 General Exam Policies Exam 2 will be in class next Tuesday, April 14, starting at 9:00 sharp relevant material: in first place topics we covered since last exam, but of course should feel com- fortable with densities, probabilities and other concepts from the first third of the semester more text problems than on problem sets, but less tedious calculations will hand out normal probability tables with exam, so don’t have to bring your own essentially same format as in first exam bring calculators closed books, closed notes have about 85 minutes to do exam I’ll give partial credit, so try to get started with all problems 2 Review 2.1 Functions of Random Variables General set-up: know p.d.f. of X , f X ( x ) (discrete or continuous) Y is a known function of X , Y = u ( X ) interested in finding p.d.f. f Y ( y ) The way how we obtain the p.d.f. f Y ( y ) depends on whether X is continuous or discrete, and whether the function u ( ) is one-to-one. Three methods · 1. if X discrete: f Y ( y ) = f X ( x ) { x : u ( x )= y } 1
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2. 2-step method if X continuous: Step 1: obtain c.d.f. F Y ( y ) F Y ( y ) = P ( u ( X ) y ) = f X ( x ) dx { x : u ( x ) y } Step 2: differentiate c.d.f. in order to obtain p.d.f.: d f Y ( y ) = F Y ( y ) dy 3. change of variables formula if (a) X continuous, and (b) u ( ) is one-to-one: · d f Y ( y ) = f X ( s ( y )) s ( y ) dy A few important examples which we discussed were: Convolution Formula: if X and Y are independent, then Z = X + Y has p.d.f. f Z ( z ) = f Y ( z w ) f X ( w ) dw −∞ Note: if densities of X and/or Y are zero somewhere, be careful with integration limits! Integral Transformation: if X continuous, then the random variable Y = F X ( X ), where F X ( · ) is the c.d.f. of X is uniformly distributed. Order Statistics: if X 1 , . . . , X n i.i.d., then k th lowest value Y k has p.d.f. f Y k ( y ) = k n F X ( y ) k 1 (1 F X ( y )) n k f X ( y ) k 2.2 Expectations 2.2.1 Expectation Definition of expectation of X if X discrete, E [ X ] = xf X ( x ) x if X continuous, E [ X ] = xf X ( x ) dx −∞ Important properties of expectations 1. for constant a , E [ a ] = a 2. for linear function of X , Y = aX + b , E [ Y ] = a E [ X ] + b 2
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3. for 2 or more random variables, E [ a 1 X 1 + . . . + a n X n + b ] = a 1 E [ X 1 ] + . . . + a n E [ X n ] + b 4. if X and Y are independent , then E [ XY ] = E [ X ] E [ Y ] expectation is measure of location of distribution of X . expectation of function Y = u ( X ) (discrete case: replace integral with sum) E [ Y ] = u ( x ) f X ( x ) dx −∞ Jensen’s Inequality: if u ( · ) is convex , then E [ u ( X )] u ( E [ X ]) 2.2.2 Variance Defined as Var( X ) = E ( X E [ X ]) 2 Measure of dispersion of X .
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