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 frst place topics we covered since last exam, but oF course should Feel com- Fortable with densities, probabilities and other concepts From the frst 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 frst 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 fnding 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
± ± ± ± ² ³ ´ 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 De±nition 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
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MIT14_30s09_lec16 - MIT OpenCourseWare http/ocw.mit.edu...

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