Hery Sar ad Joh W. Woods, robably, Sascs, ad Radom ables for geers, h ed., earso ducao Ic., 0. ISBN: 978-0-3-33-6 Chaer 6 Sascs: ar arameer smao Secos 6. Iroduco 30 Ideede, Idecally Dsrbued (..d.) Observaos 3 smao of robables 33 6. smaors 36 6.3 smao of he Mea 38 roeres of he Mea-smaor Fuco (MF) 39 rocedure for Geg a δ-cofdece Ierval o he Mea of a Normal Radom able Whe σ Is Kow 35 Cofdece Ierval for he Mea of a Normal Dsrbuo Whe σ Is No Kow 35 rocedure for Geg a δ-cofdece Ierval Based o Observaos o he Mea of a Normal Radom able whe σ Is No Kow 355 Ierreao of he Cofdece Ierval 355 6. smao of he ace ad Covarace 355 Cofdece Ierval for he ace of a Normal Radom varable 357 smag he Sadard Devao Drecly 359 smag he covarace 360 6.5 Smulaeous smao of Mea ad ace 36 6.6 smao of No-Gaussa arameers from Large Samles 363 6.7 Mamum Lelhood smaors 365 6.8 Orderg, more o erceles, aramerc Versus Noaramerc Sascs 369 The Meda of a oulao Versus Is Mea 37 aramerc versus Noaramerc Sascs 37 Cofdece Ierval o he ercele 373 Cofdece Ierval for he Meda Whe Is Large 375 6.9 smao of Vecor Meas ad Covarace Marces 376 smao of μ 377 smao of he covarace K 378 6.0 Lear smao of Vecor arameers 380 Summary 38 roblems 38 Refereces 388 Addoal Readg 389 Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 of 37 C 3800
6. Iroduco Sascs Defo: The scece of assemblg, classfyg, abulag, ad aalyzg daa or facs: Descrve sascs he collecg, groug ad reseg daa a way ha ca be easly udersood or assmlaed. Iducve sascs or sascal ferece use daa o draw coclusos abou or esmae arameers of he evrome from whch he daa came from. Theorecal Areas: Samlg Theory smao Theory Hyohess Tesg Curve fg ad regresso Aalyss of ace selecg samles from a colleco of daa ha s oo large o be eamed comleely. cocered wh mag esmaes or redcos based o he daa ha are avalable. aems o decde whch of wo or more hyoheses abou he daa are rue. aem o fd mahemacal eressos ha bes rerese he daa. (Show Cha. ) aem o assess he sgfcace of varaos he daa ad he relao of hese varaces o he hyscal suaos from whch he daa arose. (Moder erm ANOVA) We wll focus o arameer esmao (Cha. 6) ad hyohess esg (Cha. 7) Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 of 37 C 3800
Samlg Theory The Samle Mea How may samles are requred o fd a rereseave samle se ha rovdes cofdece he resuls? Defec esg, oo olls, feco raes, ec. Defos oulao: Samle: Samle Mea: he colleco of daa beg suded N s he sze of he oulao a radom samle s he ar of he oulao seleced all members of he oulao mus be equally lely o be seleced! s he sze of he samle he average of he umercal values ha mae of he samle oulao: N Samle se: S,,,,, 3 5 Samle Mea To geeralze, descrbe he sascal roeres of arbrary radom samles raher ha hose of ay arcular samle. Samle Mea, where are radom varables wh a df. Noce ha for a df, he rue mea,, ca be comue whle for a samle daa se he above samle mea, s comued. Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 3 of 37 C 3800
Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 of 37 C 3800 As may be oed, he samle mea s a combao of radom varables ad, herefore, ca also be cosdered a radom varable. As a resul, he hoed for resul ca be derved as: If ad whe hs s rue, he esmae s sad o be a ubased esmae. Though he samle mea may be ubased, he samle mea may sll o rovde a good esmae. Wha s he varace of he comuao of he samle mea? ace of he samle mea (he mea self, o he value of ) You would eec he samle mea o have some varace abou he robablsc or acual mea; herefore, s also desrable o ow somehg abou he flucuaos aroud he mea. As a resul, comuao of he varace of he samle mea s desred. For N>> or N fy (or eve a ow df), usg he colleced samles based o he ror defo of varace, a sascal esmae of he d mome ad he square of he mea. For deede (measuremes are deede of each oher) for for,,
Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 5 of 37 C 3800 As a resul we ca defe wo summao where = ad <>,, where s he rue varace (robablsc) of he radom varable,. Therefore, as aroaches fy, hs varace he samle mea esmae goes o zero! Thus a larger samle sze leads o a beer esmae of he oulao mea. Noe: hs varace s develoed based o samlg wh relaceme. Whe based o samlg whou relaceme Desrucve esg or samlg whou relaceme a fe oulao resuls aoher eresso: N N Noe ha whe all he samles are esed (N=) he varace ecessarly goes o 0. Ad all he samles have bee removed from he oulao?! The varace he mea bewee he oulao ad he samle se mus be zero as he ere oulao has bee measured!
amle: How may samles of a fely log me waveform would be requred o sure he mea s wh % of he rue (robablsc) mea value? For hs relaosh, le 0.0 0.00 Ife se, herefore assume ha you use he wh relaceme equao : Assume ha he rue meas s 0 ad ha he rue varace s 9 so ha he mea =/- a sadard devao would be 0 3. The, 9 0.00 9 0. 0. 0 900 A very large samle se sze o esmae he mea wh he % desred boud! Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 6 of 37 C 3800
Ceral Lm Theorem smae Thg of he characerzao afer usg a very large umber of samles Usg he ceral lm heorem (assume a Gaussa dsrbuo) o esmae he robably ha he mea s wh a rescrbed varace (% from he revous eamle): r 9.9 0. F 0. F 9.9 Assume ha he sascal measureme desy fuco has become Gaussa ceered aroud 0 wh a % of he mea sadard devao (assumg ha 0 ad 0. ). We ca use Gaussa/Normal Tables o deerme he robably r 0.0 9.9 0 9.9 0. r 0. 0. 9.9 0. r9.9 0. 0.83 0. 686 Ths mles ha, afer ag so may measureme o form a esmae, here s a 68.3% chace he esmae s wh % of he mea ha here s a -0.686 or 3.7% robably ha he esmae of he oulao mea s more ha % away from he rue oulao mea. or Summary, as he umber of samle measuremes creases, he desy fuco of he esmaed mea abou he rue (robablsc) mea aes o a Gaussa characersc. (based o he ceral lm heorem) Based o he varace of he samle mea comuao (relaed o umber of samles) he robably ha he measureme mea mach he robablsc mea has ow robably (based o Gaussa sascs). We wll be dealg wh Gaussa/Normal Dsrbuos as large sum szes wh some radom varable assocao haves o desy fucos ha are Gaussa Ceral Lm Theorem. Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 7 of 37 C 3800
amle #: A smaller samle sze oulao: 00 rassors Fd he mea value of he curre ga,. Assume ha: he rue oulao mea s 0 ad he rue oulao varace s 5. How large a samle s requred o oba a samle mea ha has a sadard devao of % of he rue mea? Therefore, we wa 0.00.. A smaller samle sze, samle mea varace ca be comued as N N Deermg he umber of samles eeded o mee olerace 5 00. 00.99 00 5 00. 99 5 00 6.70.9 5 A rule-of-humb s offered o defe large vs. small samle szes, he hreshold gve s 30. The ulmae goal s o have eough samles o acheve a ear-gaussa robably dsrbuo. Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 8 of 37 C 3800
Samlg Theory The Samle ace Whe dealg wh robably, boh he mea ad varace rovde valuable formao abou he DC ad AC oerag codos (abou wha value s eeced) ad he varace ( erms of ower or squared value) abou he oerag o. Therefore, we are also eresed he samle varace as comared o he rue daa varace. The samle varace of he oulao (sdev) s defed as: S ad coug ul (show he comg ages) S where s he rue varace of he radom varable. Noe: he samle varace s o equal o he rue varace; s a based esmae! To creae a ubased esmaor, scale by he basg facor o comue (sdev): S S Whe he oulao s o large, he based esmae becomes N S N ad he ubased esmae s N S S N Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 9 of 37 C 3800
Addoal oes: MATLAB ad MS cel Smulao ad sascal sofware acages allow for eher based or ubased comuaos. I MS cel here are wo dsc fucos sdev ad sdev. sdev uses (-) - h://offce.mcrosof.com/e-us/ecel-hel/sdev-fuco-h00335660.as sdev uses () - hs://suor.offce.com/e-us/arcle/stdv-fuco-f7cc88-bc-- 8-9F7DC8BB95 I MATLAB, here s a addoal flag assocae wh he sd fuco. sd sd var, flag mled as 0 var,,, flag secfed as >> hel sd sd Sadard devao. For vecors, Y = sd() reurs he sadard devao. For marces, Y s a row vecor coag he sadard devao of each colum. For N-D arrays, sd oeraes alog he frs o-sgleo dmeso of. sd ormalzes Y by (N-), where N s he samle sze. Ths s he sqr of a ubased esmaor of he varace of he oulao from whch s draw, as log as cosss of deede, decally dsrbued samles. Y = sd(,) ormalzes by N ad roduces he square roo of he secod mome of he samle abou s mea. sd(,0) s he same as sd(). Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 0 of 37 C 3800
Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 of 37 C 3800 Samlg Theory The Samle ace - roof The samle varace of he oulao s defed as S S Deermg he eeced value S S S S S S, S 3 3 S
Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 of 37 C 3800 S S S Therefore, S To creae a ubased esmaor, scale by he (u-) basg facor o comue: S S ace of he varace As before, he varace of he varace ca be comued. (Isead of dervg he values, s gve.) I s defed as S where s he fourh ceral mome of he oulao ad s defed by roof for era cred homewor cred? For he ubased varace, he resul s S S
amle: he radom me samles roblem (frs eamle) revously used where he rue meas s 0 ad ha he rue varace s 9. The, 9 9 900 ad for =900 0. 0 S for a Gaussa radom varable, he h ceral mome s 3. Therefore 3 S S 900 9 900 5800 0.80 8080 S 0. 7 The ace esmae would he be 00 S or wh S %.7% Whle 900 was seleced o rovde a mea esmae ha was wh %, he varace esmae s o early as close a.7%. More samles are requred o mrove he varace esmae. 9 Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 3 of 37 C 3800
Sascal Mea ad ace Summary For ag samles ad esmag he mea ad varace Mea ace (based) The smae A ubased esmae S A based esmae S ace of smae S ace (ubased) S S A ubased esmae S S S Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 of 37 C 3800
Bouds o he esmaes Usg he Chebyshev Iequaly Boudg he esmaed mea value f we le Therefore, for ay value lambda lm 0 lm lm 0 The robably ha he esmaed (sascal) mea s dffere from he robablsc mea s zero! Therefore he wo mus be decal for he fe samle case! Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 5 of 37 C 3800
Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 6 of 37 C 3800 Buldg a cofdece erval From Chaer, he dscree dervao saed Le be a arbrary R.V. wh ow mea ad varace. The for ay 0 Usg he Chebyshev Iequaly Dervao d f The d f d f ad d f d f Resuls #: I may be covee o defe he dela fuco erms of a mulle of he varace. Flg he bouds o he equaly adg he absolue value ad addg he mea A cofdece erval for he sascal average becomes
If we assume ha he mea value has a Gaussa dsrbuo, a eac value ca be comued for hs robably A cofdece erval ofe chose s 95% or 0.95. 0.95 0.95 0. 975.96 ad.96.96 0.95 Usg amle 6.3 effec of samle sze o he esmaed mea If he acual mea s 0 ad acual sadard devao s 3, wha are he 95% cofdece bouds? For 0 ad 3 For a wo-sdes erval 0.95 0.95 0. 975 ad =.96 0.96.96 0.95 3 5.88 5.88 0.95 If we were hog o be wh +/-0., how may samles are eeded? 0. 0. 0. 95 If ad oly f 5.88 0. or 58.8 or 357. If we oly used =6 samles, he robably of beg wh he 95% erval s??? Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 7 of 37 C 3800
5.88 5.88??? 6 6 0.735 0.735??? 0.735 0.735 0.735 0.63 0. 6 We could c a dffere cofdece erval say 50%. 0.50 0.50 0. 75 0.67 ad 0 0.67 0.67 0.50 3 0.50 If we oly used =6 samles, 0.5 0.5 0. 50 Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 8 of 37 C 3800
Gaussa Cofdece Iervals (CI): For a wo sded cofdece erval, we wa C. I. (A: eboo ses & ) Fd he arorae value for he cofdece erval seleced. (B: eboo se 3 & ) Based o he ow Gaussa mea ad varace for he esmaed R.V ad he ow umber of samles, Comue he bouds o he equaly. Alerae soluo yy. If you ow he bouds o he robably equaly ad he Gaussa sascs, comue he umber of samles eeded. boud bouds for C. I. Summary for usg Gaussa C.I (ow mea ad varace) bouds Comue wha you eed bouds C. I. Loog a he umbers The hgher he cofdece erval, he wder are he bouds. The gher are he bouds, he smaller he cofdece erval. Gaussa cofdece +/- oe sadard devao 68.3% +/- wo sadard devao 95.% +/- hree sadard devao 99.7% 90% =+/-.6 sadard devao 95% =+/-.96 sadard devao 99% =+/-.58 sadard devao Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 9 of 37 C 3800
More Gaussa Cofdece Ierval ( %) Two Tal Bouds or z : z z z 99.99% 0.005% o 99.995% 3.89 99.9% 0.05% o 99.95% 3.9 99% 0.5% o 99.5%.58 95%.5% o 97.5%.96 90% 5% o 95%.6 80% 0% ro 90%.8 50% 5% o 75% 0.675 Gaussa q values 0. c c c 0.35 0.3 q= 50.00%, =0.67 0.5 f() 0. 0.5 0. 0.05 q= 90.00%, =.65 q= 95.00%, =.960 q= 99.00%, =.576 0-5 - -3 - - 0 3 5 see Sec Gaussa.m There are oe-sded bouds ha we have o dscussed. For a Gaussa R.V. z c q for z c z bouds C. I. Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 0 of 37 C 3800
Cofdece Iervals whe we do o ow he acual varace We have he sascally comued, o-based varace esmae. Defe he ew esmaed radom varable mea fuco: as T S ad defe T To smlfy he eboo descro volvg he ch-squared dsrbuo, hs s he bass for Sude s -dsrbuo wh - degrees of freedom. The Sude s robably desy fuco (leg v=-, he degrees of freedom) s defed as where ft s he gamma fuco. v v v v v The gamma fuco ca be comued as! for ay for a eger ad () Noe ha whe evaluag he Sude s -desy fuco, all argumes of he gamma fuco are egers or a eger lus ½. () Noe ha: The dsrbuo deeds o ν, bu o μ or σ; he lac of deedece o μ ad σ s wha maes he -dsrbuo mora boh heory ad racce. Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 of 37 C 3800
Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 of 37 C 3800 h://e.weda.org/w/sude's_-dsrbuo Sude's dsrbuo arses whe (as early all raccal sascal wor) he oulao sadard devao s uow ad has o be esmaed from he daa. Noe ha: The dsrbuo deeds o ν = -, bu o μ or σ; he lac of deedece o μ ad σ s wha maes he -dsrbuo mora boh heory ad racce. T-dsrbuo cofdece erval For a wo sded cofdece erval, we wa.. I C T T T (A: eboo ses & ) Fd he arorae value based o he value v=- for he cofdece erval seleced. (H. he ables says ad, bu you are loog u v=- (o ) ad fdg = based o F T ) (B: eboo se 3 & ) Based o he ow comued varace for he esmaed R.V ad he ow umber of samles. Comue he bouds o he equaly. C I. Or he bouds o he rue mea, based o he cofdece erval are C I. c T c T T F F d f CI c c 00 for c c, -sded There are oe-sded bouds ha we have o dscussed. For T-dsrbuo R.V. c T T F d f CI c 00 for c, rgh-al C I.
Comarg he desy fucos: Sude s ad Gaussa 0. 0.35 0.3 Sudes ad Gaussa Deses Gaussa T w/ v= T w/ v= T w/ v=8 Desy fuco F T () 0.5 0. 0.5 0. 0.05 0 - -3 - - 0 3 See SudesT_lo.m ad fuco sudes_.m Sude s ft v v v v v Gaussa f e Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 3 of 37 C 3800
HW -. A very large oulao of bolar rassors has a curre ga wh a mea value of 0 ad a sadard devao of 0. The value of curre ga may be assumed o be deede Gaussa radom varables. a) Fd he cofdece lms for a cofdece level of 90% o he samle mea f s comued from a samle sze of 50. Two sded es a 90% meas ha =.65. 0.65.33 50 0.33 0.33 b) Reea ar (a) f he samle sze s. Two sded es a 90% meas ha =.65. 0.65 3.590 0 3.590 0 3.590 Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 of 37 C 3800
HW -.3 Reea roblem -. for a oe-sded cofdece erval. Resag he roblem Fd he value of he curre ga above whch 90% of he samle meas would le. (a) 50 samle sze Oe sded es a 90% meas ha Therefore, =.86 ad 0.9 or Q 0. 9 0.86.07 50 8.95 (b) samle sze Oe sded es a 90% meas, =.86 ad 0.86.799 7.0 Oe Tal Bouds Cofdece Ierval ( %) or z : z z z amles of use: 99.99% 99.99% 3.790 99.9% 99.9% 3.090 99% 99%.363 95% 95%.69 90% 90%.86 80% 80% 0.86 75% 75% 0.675 50% 50% 0 Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 5 of 37 C 3800 c c c
ercse -. A very large oulao of ressor values has a rue mea of 00 ohms ad a samle sadard devao of ohms. Fd he cofdece erval o he samle mea for a cofdece level of 95% f s comued from: a) a samle sze of 00. v = 99 Usg v=60 (o 00 gve) ad F=0.975 ( sded es) o. G-, =.00. Therefore S S S.00 0.8 00 00 0.8 00 0.8 99. 00.8 Usg v=0 (o 00 gve) ad F=0.975 ( sded es) o. G-, =.98. S.98 0.79 00 99.08 00.79 b) a samle sze of 9. v = 8 Usg v=8 ad F=0.975 ( sded es) o. G-, =.306. Therefore S S S.306 3.075 9 00 3.075 00 3.075 Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 6 of 37 C 3800
HW -. A very large oulao of bolar rassors has a curre ga wh a mea value of 0 ad a sadard devao of 0, The value of curre ga may be assumed o be deede Gaussa radom varables. b) Reea ar (a) f he samle sze s. Two sded es a 90% meas ha =.65. 0.65 3.590 0 3.590 0 3.590 If he varace was a esmaed varace sead of a ow varace. v = 0 Usg v=0 ad F=0.95 ( sded es) o. G-, =.75. Therefore S S 0 S.75 3.76 0 3.76 0 3.76 Noce ha usg a esmae varace resuls a greaer rage of values (dffereces he desy fucos). Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 7 of 37 C 3800
Sll 7- A cereal vedor s qualy corol dearme has us esed a radom samle of 0 0 ouce boes of Oa Flaes by weghg hem order o see f her 0 ouce clam s o be beleved. Ther reor, o be forwarded o maageme, mus clude a 95% cofdece erval as o he oulao mea. a) Fd he ubased mea ad sadard devao b) Deerme he 95% cofdece erval of he mea (by usg he Sude s- able). c) I geeral, f he cofdece erval becomes gher (smaller), would he cofdece level crease or decrease? Measureme Daa: 9, 8,,, 8,, 7, 9, 0, ad 7. a) Samle Mea 0, where are radom varables wh a df. 9 0 9 8 8 7 9 0 7 9. Ubased varace S 7.6 S 0...8.8..8. 0..8. 3. 067 9 S 7.6 9 3.067.75 v = 9 Usg v=9 ad F=0.975 ( sded es) o. G-, =.6. Therefore S S.75 S.6.5 0 9..5 9..5 7.98 0.5 (c) As he cofdece erval becomes gher (smaller) [% gog dow! ], he cofdece level/erval decreases. 9 Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 8 of 37 C 3800
6. Defos smaor: a fuco of he observaos vecor ha esmaes a arcular arameer. Ubased esmaor a esmaor s ubased f he esmae coverges o he correc value. Based esmaor Lear esmao Cosse a esmaor may be based. Beg covergg o a offse or ga adused value. he esmae s a lear combao of he samle os b H he esmaor s cosse f he esmae coverges o he arorae value as he umber of samles goes o fy. There are esmaors ha mmze he varace he esmae from he samle values. There are esmaors ha mmze he mea-squared error for he samle values. Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 9 of 37 C 3800
Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 30 of 37 C 3800 6.6 smao of o-gaussa arameers. Usg he Chebyshev Iequaly The cofdece erval for he sascal average became Bu based o he ceral lm heorem, we deermed he robably o be relaed as a sum RV where he resulg RV becomes Gaussa, wh rescrbed meas ad varaces based o he orgal desy fucos. or Whe he al dsrbuos of he summed radom varables are o-gaussa, he mea ad varace may be relaed, for eamle he eoeal dsrbuo. amle 6.6 eoeal dsrbuo u e f where ad We wsh o esmae bouds for lambda
amle 6.6 eoeal dsrbuo Deerme he 95% cofdece erval for 6 samles whe he daa esmaed mea s 3.5. The, 3.5.96.96.96 6 3.5 0.755.5 0.95 3.5 3.5 0.6 0.356 0. 95 0.95 6 Noe ha he esmaed s based a mea of 3.5, 0.86 amle 6.6 3 Beroull dsrbuo mf q, for q m q VAR The sascal summao based o CLT should rovde The rage of he bouds become q m ad S q q Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 3 of 37 C 3800
Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 3 of 37 C 3800 q q Deermg he erval bouds ad wdh solve for erms of all he oher varables! Remember ha q=- q Comlee he squared erm o he lef Smlfy as bes ossble, ad The dsace bewee he wo soluos goes o zero as creases Noe: I have o clue wha he eboo dd
amle 6.6 Is a far co? If we ge 7 heads afer ossg a co 00 mes, s far wh a 95% cofdece erval? 0.5 0.5 0.5 0.5 0. 95 00 00.96 0.5 0.5.96.96 0.95 0 0 0.098 0.098 0. 95 0.5 0.098 0.5 0.098 0. 95 0.0 0.598 0. 95 Therefore, 0.7 s wh he acceable rage. Aleraely 0.5 0.5 00 0.5 0.5 0.95 00 0.7 0.098 0.7 0.098 0. 95 0.37 0.568 0. 95 If he umber of co fls were 00 wh he same rooroal resuls 0.5 0.5.96.96 0.95 3.6 3.6 Aleraely 0.083 0.083 0. 95 0.77 0.583 0. 95 (0.7 s o he rage) 0.7 0.983 0. 95 (0.5 s o he rage) We would have o say he co s based. The values are o wh he 95% cofdece ervals! Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 33 of 37 C 3800
6.7 Mamum Lelhood smaors The lelhood fuco ca be roerly defed as L ;,,, f We are eresed fdg he value of hea,, ha mamzes hs fuco! To solve ae he dervave, se o zero, solve ad deerme he mma ad mama. c he global mama! asy rgh?! amle 6.7 Beroull RV of a uow robably. Wha s he mamum lelhood esmae of f afer flg cos mes, we have heads? r, for 0,,,, We defe a lelhood fuco Deerme he dervave d r Y d r Y 0 0 0 0 The roos are a 0,, Two are mma (=0 ad =), herefore he ML robably s ML Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 3 of 37 C 3800
Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 35 of 37 C 3800 amle 6.7 Deerme he mea of a Gaussa R.V wh ow varace. For e m f The o robably lelhood esmae for samle rals becomes. e f L A smlfcao ofe erformed s o use he log-lelhood fuco or f f L log log log e log log log L L e log log log L log log log Tag he dervave 0 log log d d d d L d d 0 ML
Mamum Lelhood smaor roeres Ivarace: f a ML s foud for hea,, he he ML of a fuco of hea s he fuco of he ML. For The ML y h has y y ML h ML from hs://e.weda.org/w/mamum_lelhood Cossecy: he sequece of MLs coverges robably o he value beg esmaed. Asymoc ormaly: as he samle sze creases, he dsrbuo of he ML eds o he Gaussa dsrbuo wh mea \hea ad covarace mar equal o he verse of he Fsher formao mar. ffcecy,.e., acheves he Cramér Rao lower boud whe he samle sze eds o fy. Ths meas ha o cosse esmaor has lower asymoc mea squared error ha he ML (or oher esmaors aag hs boud). Secod-order effcecy afer correco for bas. Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 36 of 37 C 3800
6.8 Orderg, Rag ad erceles ercele: hs://e.weda.org/w/ercele A ercele (or a cele) s a measure used sascs dcag he value below whch a gve erceage of observaos a grou of observaos fall. For eamle, he 0h ercele s he value (or score) below whch 0 erce of he observaos may be foud. Teboo: The u-h ercele of s he umber u such ha F ( u )=u. u F u Oe would say ha he resul u s he u h ercele. amle 6.8- Assume a erso s IQ s dsrbued as N(00,00). Tha s, a Gaussa ormal wh mea of 00, a varace of 00 ad a sadard devao of 0. The a IQ of 5 would be defed a wha ercele of he oolaos? The dvdual s he 93 rd ercele for IQ. z 500.5 0.5 0. 933 Meda: The meda of a oulao s defed where half of he oulao s above ad half s below he value. F meda 0. 5 For some of he dsrbuos descrbed, he mea ad he meda are o equal! For eamle, he eoeal dsrbuo. l 0.5 0.69 meda whereas Noes ad fgures are based o or ae from maerals he course eboo: robably, Sascs ad Radom rocesses for geers, h ed., Hery Sar ad Joh W. Woods, earso ducao, Ic., 0. B.J. Bazu, Fall 06 37 of 37 C 3800