Goals. Economics To review items used throughout the text. Appendix A: A Review of Some Statistical Concepts

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1 Goals Ecoomcs 337 Apped A: A Revew of Some Statstcal Cocepts To revew tems used throughout the tet. Mathematcal operators Statstcal cocepts Ths wll establsh a laguage to help us troduce ew materal Apped A A. Summato & Product Operators A SIGMA dcates summato = X = X + X X A PI dcates multplcato = X = X X... X A. Sample Space, Pots, ad Evets Populato (Sample Space) Set of all possble outcomes of a radom epermet Every elemet of the sample space s called a Sample Pot Eample: Flppg cos Sample space cossts of 4 sample pots {HH, HT, TH, TT} Apped A 3 Apped A 4 A. Sample Space, Pots, ad Evets A Evet s a subset of the sample space Let A deote head ad tal A s a evet - {HT, TH} belog to A A evet s mutually eclusve f the occurrece of a evet precludes aother If HH occurs, HT caot A. Sample Space, Pots, ad Evets Evets are (collectvely) ehaustve f they cover all possble epermet outcomes {HH}, {TT}, ad A are ehaustve evets f you flp cos you are boud to get oe of these three evets Apped A 5 Apped A 6

2 A.3 Probablty ad Radom Varables The probablty of evet A, P(A), s the proporto of tmes the evet wll occur repeated trals. Propertes 0 PA ( ), for every A If A, B, C,... are ehaustve sets, the PA ( + B+ C+...) = If A, B, C,... are mutually eclusve, the PA ( + B+ C+...) = PA ( ) + PB ( ) + PC ( ) +... A.3 Probablty ad Radom Varables Epermet: throwg de Sample space cossts of 6 evets {,, 3, 4, 5, 6} Probablty of ay evet s /6 P( ) = Apped A 7 Apped A 8 A.3 Probablty ad Radom Varables A Radom Varable (rv) s determed by the outcome of a chace epermet RVs are deoted by captal letters (X,Y,Z) Values take by them are deoted by (,y,z) Eample: X = the outcome of throwg de = (value of a sgle outcome) A.3 Probablty ad Radom Varables A RV s dscrete f t takes o a fte (or coutably fte) umber of values The sum of de (, 3, 4, 5,, ) A RV s cotuous f t ca take ay value wth some terval Heght of a dvdual, temp. Depeds o accuracy of measuremet (you ca always go to smaller fractos) Apped A 9 Apped A 0 For dscrete RVs f ( ) = P( X = ) for =,,...,,... f( ) = 0 for For cotuous RVs b ( ) = ; ( ) = ( ) f d f d Pa b a Eample: roll de = f () = Apped A Apped A

3 Dscrete Jot PDF Probablty that RVs X ad Y both ht a partcular umber (, ) = ( = ad = ) f y P X Y y = 0 whe X ad Y y Margal PDF Probablty that oe RV equals a partcular umber (regardless of the other RV) ( ) = ( ) y ( ) = ( ) f f, y margal PDF of X f y f, y margal PDF of Y Apped A 3 Apped A 4 The Codtoal PDF of X gves the probablty that X = gve that Y = y. ( ) = ( = = ) f (, y) ( ) = f ( y) f y PX Y y f y Eample: Jot PDF of dscrete varables X ad Y Y=3 Y=6 X= X= X= X= Apped A 5 Apped A 6 Margal PDF of = - s gve by ( ) f ( y) f = =, = = 0.7 y Codtoal PDF of = - gve y = 3 s gve by ( = y= ) f ( y= ) f, f ( = y= 3) = = = A.5 Characterstcs of The characterstcs of a Probablty Dstrbuto are called momets Frst momet: Mea (epected value) ( ) ( ) E X = f = μ Apped A 7 Apped A 8 3

4 A.5 Characterstcs of Epected value of throwg two de 3 EX ( ) = = A.5 Characterstcs of The varace of X ( d momet) measures the spread of the values of X aroud ts epected value var X = E X μ ( ) ( ) or E( X ) μ E( X ) E( X) ( X μ) f ( ) = = = Apped A 9 Apped A 0 A.5 Characterstcs of Varace of throwg de var ( X ) = ( 7) + ( 3 7 ) ( 7) = A.5 Characterstcs of The postve square root of the varace s the stadard devato How the dvdual X values are spread aroud the mea How RVs vary together s the covarace ( X Y) = E{ ( X )( Y )} = E( XY) = ( X μ)( Y μy) f (, y) cov, y μ μ μ μ y y Apped A Apped A A.6 Some Importat Theoretcal The Normal Dstrbuto The ormal PDF of a cotuous rv X s gve by ( μ) f ( ) = ep, < < σ π σ A.6 Some Importat Theoretcal The Normal Dstrbuto Gve mea & varace, oe ca fd the probablty that X wll le a certa terval by tegratg the PDF. To make thgs smpler, we trasform the rv X to a stadardzed ormal varable Z μ Z= σ Apped A 3 Apped A 4 4

5 Start to Pcture Data They ormalze to the same thg! Apped A 5 Apped A 6 A.6 Some Importat Theoretcal The Normal Dstrbuto Ths gves a rv Z wth zero mea ad ut varace f Z π Z ( ) = ep The probabltes ca ow be looked up the stadardzed ormal table (D.) A.6 Some Importat Theoretcal Eample: ( 8, 4) X N What s the probablty that X wll be betwee 4 ad? X 4 8 Z Z μ = = = σ X μ 8 = = = σ ( ) ( ) P Z = P 0 Z = Apped A 7 Apped A 8 Pcturg data aga A.7 Statstcal Iferece: Estmato Suppose we have a rv, X, wth ukow mea Wth a radom sample of sze, the mea ca be estmated ˆ μ = = = X Apped A 9 Apped A 30 5

6 A.7 Statstcal Iferece: Estmato We ca also estmate a terval The cofdece that a estmate les wth a partcular rage (cofdece terval) θ = true value θˆ ˆ, θ = two estmates Pr θ θ θ = α, 0 < α < ( ˆ ˆ ) Ths s a 00(- α)% Cofdece terval A.8 Statstcal Iferece: Hypothess Testg Testg hypotheses leds support to our estmatos Suppose we have a rv X wth a kow PDF, but ukow parameters (.e. mea) PDF = f ( ; θ ), ukow θ sample sze θˆ (a estmate) Questo: s θˆ 'compatble' wth some hypotheszed value θ *? Apped A 3 Apped A 3 A.8 Statstcal Iferece: Hypothess Testg Null (mataed) hypothess: H 0 : θ = θ * Alteratve hypothess: H : θ θ * We the decde f the sample accepts or rejects the ull usg ether a cofdece terval approach, or test of sgfcace. Why s ths stuff mportat? For EVERYTHING we do, we wll be uder the assumpto that we draw a sample (data set) that wll accurately represet the etre populato.e. sample mea represets actual mea, Apped A 33 Apped A 34 Why? Because our goal s ot to determe what happeed the past, but to use past formato to help us fgure out what wll happe the future. Apped A 35 6