Random variables and sampling theory

Transcription

1 Revew Rdom vrbles d smplg theory [Note: Beg your study of ths chpter by redg the Overvew secto below. The red the correspodg chpter the textbook, vew the correspodg sldeshows o the webste, d do the strred exercses. Do the ddtol exercses the subject gude. Check the swers to the exercses. Flly, check tht you hve tted ll the lerg outcomes lsted below.] Overvew The textbook d ths gude ssume tht you hve prevously studed bsc sttstcl theory d hve soud uderstdg of the followg topcs: descrptve sttstcs (me, med, qurtle, vrce, etc.) rdom vrbles d probblty smplg theory d estmto propertes of estmtors (ubsedess, effcecy, d cosstecy) orml dstrbuto hypothess testg, cludg: t tests Type I d Type II error the sgfcce level d power of t test oe-sded versus two-sded t tests. You do ot eed to hve studed smplg wthout replcemet, the Posso dstrbuto, or lyss of vrce. There re my excellet sttstcs textbooks. The followg re lsted s exmples: Freud, J.E. () Moder Elemetry Sttstcs (teth edto) Wocott, T.H., d R.J. Wocott (99) Itroducto to Sttstcs for Busess d Ecoomcs (fourth edto) (or y smlr ttle by these uthors) The Revew chpter of my textbook s ot substtute for sttstcs course. It hs the much more lmted objectve of provdg opportuty for revsg some key sttstcl cocepts d results tht wll be used tme d tme g the course. Your prevous studes sttstcs should hve covered these topcs, but you my ot hve gve them the tteto tht they ow deserve. They re cetrl to ecoometrc lyss d f you hve ot ecoutered them before, you should postpoe your study of ecoometrcs d study sttstcs frst. Lerg outcomes After workg through the correspodg chpter the textbook, studyg the correspodg sldeshows, d dog the strred exercses the textbook d the ddtol exercses ths gude, you should be ble to expl wht s met by:..7

2 expectto populto vrce ubsedess effcecy cosstecy. Further, you should be ble to expl why these cocepts re mportt. The lst two terms re ofte msuderstood by studets tkg troductory ecoometrcs course, so mke sure tht you wre of ther precse megs. You should be ble to use the expected vlue rules d the vrce d covrce rules. Addtol exercses AR. A rdom vrble hs cotuous uform dstrbuto from to. Defe ts probblty desty fucto. probblty desty AR. Fd the expected vlue of Exercse AR., usg the expresso gve Box R. the text. AR. Derve E( ) for defed Exercse AR., usg the expresso gve Box R.. AR. Derve the populto vrce d the stdrd devto of s defed Exercse AR., usg the expresso gve Box R. AR.5 Usg equto (R.9), fd the vrce of the rdom vrble defed Exercse AR. d show tht the swer s the sme s tht obted Exercse AR.. (Note: You hve lredy clculted E() Exercse AR. d E( ) Exercse AR..) AR. A rdom vrble hs cotuous uform dstrbuto over the tervl from to θ, where θ s ukow prmeter. The followg three estmtors re used to estmte θ, gve smple of observtos o : () twce the smple me (b) the lrgest vlue of the smple (c) the sum of the lrgest d smllest vlues of the smple. Expl verblly whether or ot ech estmtor s () ubsed () cosstet. Aswers to the strred exercses the text R. A rdom vrble s defed to be the lrger of the two vlues whe two dce re throw, or the vlue f the vlues re the sme. Fd the probblty dstrbuto for. Aswer: The tble shows the possble outcomes. The probblty dstrbuto s derved by coutg the umber of tmes ech outcome occurs d dvdg by. The probbltes hve bee wrtte s frctos, but they could eqully well hve bee wrtte s decmls.

3 red gree Vlue of 5 Frequecy Probblty / / 5/ 7/ 9/ / R. Fd the expected vlue of Exercse R.. Aswer: The tble s bsed o Tble R. the textbook. It s good de to guess the outcome before dog the rthmetc. I ths cse, sce the hgher umbers hve the lrgest probbltes, the expected vlue should clerly le betwee d 5. If the clculted vlue does ot coform wth the guess, t s possble tht ths s becuse the guess ws poor. However, t my be becuse there s error the rthmetc, d ths s oe wy of ctchg such errors. p p / / / / 5/ 5/ 7/ 8/ 5 9/ 5/ / / Totl /.7 R.7 Clculte E( ) for defed Exercse R.. Aswer: The tble s bsed o Tble R. the textbook. Gve tht the lrgest vlues of hve the hghest probbltes, t s resoble to suppose tht the swer les somewhere the rge 5. The ctul fgure s.97. p p / / / / 9 5/ 5/ 7/ / 5 5 9/ 5/ / 9/ Totl 79/.97

4 R. Clculte the populto vrce d the stdrd devto of s defed Exercse R., usg the defto gve by equto (R.8). Aswer: The tble s bsed o Tble R. the textbook. I ths cse t s ot esy to mke guess. The populto vrce s.97, d the stdrd devto, ts squre root, s.. Note tht four decml plces hve bee used the workg, eve though the estmte s reported to oly two. Ths s to elmte the possblty of the estmte beg ffected by roudg error. p μ ( μ ) ( μ ) p / / / / / / Totl.975 R. Usg equto (R.9), fd the vrce of the rdom vrble defed Exercse R. d show tht the swer s the sme s tht obted Exercse R.. (Note: You hve lredy clculted μ Exercse R. d E( ) Exercse R.7.) Aswer: E( ) s.97 (Exercse R.7). E( ) s.7 (Exercse R.), so s.. Thus the vrce s The lst-dgt dscrepcy betwee ths fgure d tht Exercse R. s due to roudg error. R. Let ρ HT be the correlto betwee humdty, H, d temperture mesured degrees Fhrehet, F. Demostrte tht the correlto coeffcet s uffected f temperture s sted mesured degrees Celsus, C. Note: C 5/9 (F ). Aswer: We strt by otg tht ( C C ) / 9( F F ) 5. The μ ρ HC H )( C C ) H ) ( C C ) 5 H ) ( F F ) 5 H ) ( F F ) 9 9 H )( F F ) H ) ( F F ) ρ HT R. Show tht, whe you hve observtos, the codto tht the geerlzed estmtor (λ λ ) should be ubsed estmtor of μ s λ λ. Aswer: E(Z) E(λ λ ) E(λ ) E(λ ) λ E( ) λ E( ) λ μ λ μ (λ λ )μ

5 5 Thus E(Z) μ requres λ λ. R.9 I geerl, the vrce of the dstrbuto of estmtor decreses whe the smple sze s cresed. Is t correct to descrbe the estmtor s becomg more effcet? Aswer: No, t s correct. Whe the smple sze creses, the vrce of the estmtor decreses, d s cosequece t s more lkely to gve ccurte results. Becuse t s mprovg ths mportt sese, t s very temptg to descrbe the estmtor s becomg more effcet. But t s the wrog use of the term. Effcecy s comprtve cocept tht s used whe you re comprg two or more ltertve estmtors, ll of them beg ppled to the sme dt set wth the sme smple sze. The estmtor wth the smllest vrce s sd to be the most effcet. You cot use effcecy s suggested the questo becuse you re comprg the vrces of the sme estmtor wth dfferet smple szes. R. A rdom vrble hs ukow populto me μ d populto vrce. A smple of observtos { σ,, } s geerted. Show tht Z 8 s ubsed estmtor of μ. Show tht the vrce of Z does ot ted to zero s teds to fty d tht therefore Z s cosstet estmtor, despte beg ubsed. Aswer: The weghts sum to uty, so the estmtor s ubsed. However ts vrce s Z σ σ Ths teds to s becomes lrge, ot zero, so the estmtor s cosstet. / σ Note: the sum of geometrc progresso s gve by Hece 8 d

6 s becomes lrge. Aswers to the ddtol exercses AR. The totl re uder the fucto over the tervl [, ] must be equl to. Sce the legth of the rectgle s, ts heght must be.5. Hece f().5 for, d f().5 for < d >. AR. Obvously, sce the dstrbuto s uform, the expected vlue of s. However we wll derve ths formlly. ( ) ( ).5 d d f E AR. The expected vlue of s gve by ( ) ( )..5 d d f E AR. The vrce of s gve by [ ] ( ) [ ] ( ) [ ] ( ) [] d d d f E μ μ The stdrd devto s equl to the squre root,.577. AR.5 From Exercse AR., E( ).. From Exercse AR., the squre of E() s. Hece the vrce s., s Exercse AR.. AR. () It s evdet tht ( ) ( ). θ E E Hece s ubsed estmtor of θ. The vrce of s σ. The vrce of s therefore σ. Ths wll ted to zero s teds to fty. Thus the dstrbuto of wll collpse to spke t θ d the estmtor s cosstet. (b) The estmtor wll be bsed dowwrds sce the hghest vlue of the smple wll lwys be less th θ. However, s creses, the dstrbuto of the estmtor wll be cresgly cocetrted rrow rge just below θ. To put t formlly, the probblty of the hghest vlue

7 7 ε beg more th ε below θ wll be d ths wll ted to zero, o mtter how smll ε s, s θ teds to fty. The estmtor s therefore cosstet. It c fct be show tht the expected vlue of he estmtor s θ d ths teds to θ s becomes lrge. (c) The estmtor wll be ubsed. Cll the mxmum vlue of the smple mx d the mmum vlue m. Gve the symmetry of the dstrbuto of, the dstrbutos of mx d m wll be detcl, except tht tht of mx wll be to the rght of d tht of mx wll be to the left of θ. Hece, for y, E( ) E( ) m θ mx d the expected vlue of ther sum s equl to θ. The estmtor wll be cosstet for the sme reso s expled (b). The frst fgure shows the dstrbutos of the estmtors () d (b) for,, smples wth oly four observtos ech smple, wth θ. The secod fgure shows the dstrbutos whe the umber of observtos ech smple s equl to. The tble gves the mes d vrces of the dstrbutos s computed from the results of the smultos. If the mes squre error s used to compre the estmtors, whch should be preferred for smple sze? For smple sze? (b).5.5 () Smple sze 5 5 (b) 5 ().5.5 Smple sze

8 8 Smple sze Smple sze () (b) () (b) me vrce estmted bs estmted me squre error It c be show (Lrse d Mrx, A Itroducto to Mthemtcl Sttstcs d Its Applctos, p.8, tht estmtor (b) s bsed dowwrds by mout θ/( ) d tht ts vrce s θ /( ) ( ), whle estmtor () hs vrce θ /. How lrge does hve to be for (b) to be preferred to () usg the me squre error crtero? The crushg superorty of (b) over () my come s surprse, so ccustomed we re to fdg tht the smple me the best estmtor of prmeter. The uderlyg reso ths cse s tht we re estmtg boudry prmeter, whch, s ts me mples, defes the lmt of dstrbuto. I such cse the optml propertes of the smple me re o loger gurteed d t my be eclpsed by score sttstc such s the lrgest observto the smple. Note tht the stdrd devto of the smple me s versely proportol to, whle tht of (b) s versely proportol to (dsregrdg the dffereces betwee,, d ). (b) therefore pproches ts lmtg (symptotclly ubsed) vlue much fster th () d s sd to be supercosstet. We wll ecouter supercosstet estmtors oly oce the text, d tht s whe we come to cotegrto Chpter. Note tht f we multply (b) by ( )/, t s ubsed for fte smples s well s supercosstet.