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lecture16

Course: STAT 302, Spring 2011
School: UBC
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302, Stat Introduction to Probability Jiahua Chen January-April 2011 Jiahua Chen () Lecture 16 January-April 2011 1 / 23 Conditional distribution: continuous random variables Consider the case where X and Y have joint density function f (x , y ). Similar to the discrete case, we may attempt to compute P ( Y = y |X = x ) = P (X = x , Y = y ) . P (X = x ) Yet this is not feasible because P (X = x ) = 0....

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302, Stat Introduction to Probability Jiahua Chen January-April 2011 Jiahua Chen () Lecture 16 January-April 2011 1 / 23 Conditional distribution: continuous random variables Consider the case where X and Y have joint density function f (x , y ). Similar to the discrete case, we may attempt to compute P ( Y = y |X = x ) = P (X = x , Y = y ) . P (X = x ) Yet this is not feasible because P (X = x ) = 0. However, the notation of conditional distribution is just as desirable. Jiahua Chen () Lecture 16 January-April 2011 2 / 23 Continuous random variables Suppose fX (x ) > 0 in a neighborhood of x = a. Let a = [a, a + ], b = [b , b + ] for some > 0. It is seen that P (X a ) fX (a) > 0. Similarly, P (X a , Y b ) f (a, b )2 . Jiahua Chen () Lecture 16 January-April 2011 3 / 23 Continuous random variables Hence, P ( Y b |X a ) f (a, b ) = fX ( a ) which is is well dened. We hence dene fY | X ( y | x ) = f (x , y ) fX ( x ) as the conditional probability density function of Y given X = x . You may notice that this expression is very close to the expression of the conditional pmf in discrete case. Jiahua Chen () Lecture 16 January-April 2011 4 / 23 Example Suppose the joint pdf of X and Y is given by f (xy ) = for 0 < x < 1 and 0 < y < 1. What is the conditional pdf of X given Y = y ? What is the conditional pdf of Y given X = x ? 12 x (2 x y ) 5 Jiahua Chen () Lecture 16 January-April 2011 5 / 23 Example Recall the general form of conditional pdfs f (x , y )/fX (x ) and f (x , y )/FY (y ) we need only nd the marginal pdfs to answer these two questions. What is the conditional pdf of X given Y = y ? The marginal pdf of Y is fY (y ) = 1 x =0 2 12 x (2 x y )dx = (4 3y ). 5 5 Hence, when x (0, 1), the conditional pdf of y fY |X (y |x ) = for 0 < y < 1. 6x ( 2 x y ) 4 3y Jiahua Chen () Lecture 16 January-April 2011 6 / 23 Example Recall the general form of conditional pdfs f (x , y )/fX (x ) and f (x , y )/FY (y ) we need only nd the marginal pdfs to answer these two questions. What is the conditional pdf of X given Y = y ? The marginal pdf of Y is fY (y ) = 1 x =0 2 12 x (2 x y )dx = (4 3y ). 5 5 Hence, when x (0, 1), the conditional pdf of y fY |X (y |x ) = 6x ( 2 x y ) 4 3y for 0 < y < 1. The conditional pdf of Y given, for instance, X = 1 is not dened. The conditional pdf of Y at y (0, 1) equals 0. Jiahua Chen () Lecture 16 January-April 2011 6 / 23 Example What is the conditional pdf of Y given X = x ? The marginal pdf of X is fX (x ) = 1 y =0 12 6 x (2 x y )dy = (3x x 2 ). 5 5 Hence, for 0 < y < 1 given any X = x (0, 1), fY |X (y |x ) = 2(2 x y ) 2x ( 2 x y ) = . 3 2x 3x 2x 2 Jiahua Chen () Lecture 16 January-April 2011 7 / 23 Example: Bivariate normal Two random variables X and Y have bivariate normal distribution if their joint pdf is given by f (x , y ) = where g (x , y ) = (x x )(y y ) (y y )2 (x x )2 2 + . 2 2 x x y y 1 2x y 1 2 exp{ 1 g (x , y ) } 2( 1 2 ) Note x , y are variables in the joint pdf, while x , y are means, x , y are standard deviations of X and Y , plus is the correlation coecient. Because of above, x , y are positive, and [1, 1]. Simply, the density function is given by exp(g (x , y )) where g (x , y ) is a positive denite quadratic form in x , y . Jiahua Chen () Lecture 16 January-April 2011 8 / 23 Bivariate normal: marginal distribution Apparently, the marginal distributions of X and Y are both normal. By complete the square, we nd g (x , y ) = (x x )(y y ) (y y )2 (x x )2 2 + 2 2 x x y y = (x x ) (y y ) x y 2 + ( 1 2 ) (y y )2 2 y Jiahua Chen () Lecture 16 January-April 2011 9 / 23 Bivariate normal: marginal distribution The marginal pdf of Y is hence given by fY ( y ) = C exp{ exp{ 1 g (x , y )}dx 2( 1 2 ) =C (x x ) (y y ) 1 2) 2( 1 x y 2 (y y ) exp{ } 2 2y (y y )2 } 2 2y 2 }dx = C exp{ Note C depends parameter values only and its value changes from one line to another. However, its exact value is not important in our computation. Jiahua Chen () Lecture 16 January-April 2011 10 / 23 Bivariate normal: marginal distribution It is seen that the pdf of Y is proportional to exp{ which has to be (y y )2 } 2 2y 2 or N (y , y ). (y y )2 1 exp{ } 2 2y 2y Jiahua Chen () Lecture 16 January-April 2011 11 / 23 Bivariate normal: marginal and conditional distribution 2 Similarly, the marginal distribution of X is N (x , x ). Before we give the conditional pdf of X given Y = y , have a look again: f (x , y ) = where g (x , y ) = (x x )(y y ) (y y )2 (x x )2 2 + . 2 2 x x y y 1 2x y 1 2 exp{ 1 g (x , y ) 2( } 1 2 ) Jiahua Chen () Lecture 16 January-April 2011 12 / 23 Bivariate normal: marginal and conditional distribution and that (x x ) (y y ) g (x , y ) = x y 2 + ( 1 2 ) (y y )2 . 2 y We nd the conditional pdf of X given Y = y is fX |Y (x |y ) = C exp (x x ) (y y ) 1 2) 2( 1 x y 1 2 2x (1 2 ) 2 = C exp with ( x x |y ) 2 x |y = x + Jiahua Chen () x (y y ). y January-April 2011 13 / 23 Lecture 16 Bivariate normal: marginal and conditional distribution The form fX |Y (x |y ) = C exp 1 2 2x (1 2 ) ( x x |y ) 2 implies that X |Y = y is normally distributed with conditional mean x |y = x + and conditional variance 2 2 x |y = x (1 2 ). 2 This is a reduction from x . x (y y ). y Jiahua Chen () Lecture 16 January-April 2011 14 / 23 Bivariate normal: marginal and conditional distribution Having conditional variance 2 2 x |y = x (1 2 ). implies that knowing the value of Y is helpful to predict the observed value of X . Jiahua Chen () Lecture 16 January-April 2011 15 / 23 Conditional mean and conditional variance When X and Y are discrete, the conditional pmf of Y given X = x is given by p (x , y ) P ( Y = y |X = x ) = = pY | X ( y | x ) . pX (x ) When X and Y are continuous, the conditional pdf of Y given X = x is given by f (x , y ) = fY | X ( y | x ) . fX ( x ) Note only they are called pmf and pdf, they are indeed pmf and pdf (as function of y ). Jiahua Chen () Lecture 16 January-April 2011 16 / 23 Conditional mean and conditional variance To avoid confusion, I use X = a instead of X = x in the following. For discrete r.v.s, we have (1) 1 P (Y = y |X = a) 0 for all y . (2) y P (Y = y |X = a) = 1. For continuous r.v.s, we have f ( a ,y ) (1) fY |X (y |a) = f (a) 0 for all y . X (2) y fY |X (y |a)dy = 1. Both indicate that the conditional distribution is also distribution. Jiahua Chen () Lecture 16 January-April 2011 17 / 23 Conditional mean and conditional variance We may compute various moments of the conditional distribution: For discrete one, we have E [g ( Y ) |X = a ] = For continuous one, we have g (y )P (Y y = y |X = a ) . E [g ( Y ) |X = a ] = g (y )fY |X (y |a)dy . The conditional expectation of g (Y ) depends on the specic value a we choose for X . We usually use x instead of a for a potential value of X . Jiahua Chen () Lecture 16 January-April 2011 18 / 23 Example Let X1 and X2 be the arrival times of rst two students for my oce hour. Assume that X1 has exponential distribution with pdf ( x > 0) f1 (x ) = exp(x ) Assume that given X1 = a, the pdf of X2 is given by f2|1 (x |X1 = a) = exp((x a)). for x > a. What is their joint pdf? Jiahua Chen () Lecture 16 January-April 2011 19 / 23 Example The joint pdf is given by f ( x1 , x2 ) = f 2 | 1 ( x2 | x1 ) f 1 ( x1 ) and we must keep close track of these 1s and 2s. The answer is f (x1 , x2 ) = exp((x2 x1 )) exp(x1 ) = 2 exp(x2 ) for > x2 > x1 > 0. The range is crucial in this computation. What is the conditional pdf of X1 given X2 = b ? Jiahua Chen () Lecture 16 January-April 2011 20 / 23 Example What is the conditional pdf of X1 given X2 = b ? Let us nd the marginal pdf of X2 : for any b > 0, b f2 ( b ) = f (x1 , b )dx1 = 0 2 exp(b )dx1 = 2 b exp(b ) Do you know the name of this distribution? The conditional pdf of X1 given X2 = b is hence f 1 | 2 ( x1 | b ) = for 0 < x1 < b . 2 exp(b ) 1 = 2 b exp( b ) b Jiahua Chen () Lecture 16 January-April 2011 21 / 23 Example Keeping tracking the range is hard. You may go as follows: f 1 | 2 ( x1 | b ) = 2 exp(b )I (0 < x1 < b ) 1 = I ( 0 < x1 < b ) . 2 b exp( b )I (0 < x < b ) b 1 Note that this is a function of x1 , and b is regarded as a number. Jiahua Chen () Lecture 16 January-April 2011 22 / 23 Example Keeping tracking the range is hard. You may go as follows: f 1 | 2 ( x1 | b ) = 2 exp(b )I (0 < x1 < b ) 1 = I ( 0 < x1 < b ) . 2 b exp( b )I (0 < x < b ) b 1 Note that this is a function of x1 , and b is regarded as a number. Does it help to use b instead of x2 here? For instance, if b = 20mins , then f1|2 (x |20) = which is uniform on [0, 20]. What is the conditional expectation and variance of X1 given X2 = 20? Jiahua Chen () Lecture 16 January-April 2011 22 / 23 1 I (0 < x < 20) 20 Example Given X2 = 20 mins, then f1|2 (x |20) = Hence E (X1 |X2 = 20) = 2 E (X1 |X2 = 20) = 1 I (0 < x < 20). 20 1 20, 2 1 ?? = 202 . 3 ?? = Therefore, var(X1 |X2 = 20) = 1 12 Replace 20 by x2 yourself to repeat the computation/derivation. 202 . Jiahua Chen () Lecture 16 January-April 2011 23 / 23
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3.tmi Sastitmi: lognendingar rjrhljreglur Fleirtalanafnora,greinisoglsingarora Samrminafnoraoglsingarora Avxl dag: framumAvxl Frumlag,sagnfylling,andlagReglaumAvxl Stemvowelalteration:a~. Intwokindofcontext:saga 1)Iftheendingcontainsthevowelu.sgu
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8.tmi Sastitmi: dag: Eignarfallnafnora,lsingaroraoggreinis Notkuneignarfalls OrareignarsambndumNotkunef. 1)Inobjectpositionwithcertainverbsand certainprepositions: 2)Toindicatethepossessor(theowner)in possessiveconstructions: Jnsaknarstlkunnar Jnfert
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10.tmi Sastitmi: dag: kk.no.oglo.nendingarnf.et. Meiraumlo.nendingar(tvkv) Brottfallsrhljsrstofnitvkvrano.og lo.Tvkvlo.kk.nendingar nf.et.kk. kvk. et.nf.viturmaurviturkona gulpeysa et.nf.gulurdiskur ertilbrigiafurendingunni Sama beygingardmi og gulur,
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11. tmi Sasti tmi Brottfall srhljs r stofni tvkvra no. og lo. dag Sagnir Nt sagnaUm sagnir Sgn (ft.: sagnir) = sagnor (so.) Sagnbeyging Mismunandi endingar Innskotsstafur j milli stofns og endinga sumum myndum N srhljavxl: B-vxlNafnhttur Nafnhttu
Uni. Iceland - HUMANITIES - ÍSE102G
12. tmi Sasti tmi Nt sagna dag Meira um nt j-innskotEndingar nt et.1. 2. 3. ft.1. 2. 3. V11 - -r -r -um -i -a V2 -i -ir -ir -um -i -a V32 + S3 - -ur (-r/-/-t)4 f-r, fer-, les-t -ur (-r/-) 4 f-r, fer-, les- -um -i -aAthugasemdir 1) Heldur nh. a et.
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13. tmi Sasti tmi j-innskot dag Srhljavxl stofni: B-vxl Notkun ntarYfirlit1 kalla 2 heyra 3 telja tel- tel-ur tel-ur tel-j-um tel-j-i tel-j-a 4 brjta brt- brt-ur brt-ur 5 f 6 fara 7 lesa les- les-t les- kalla- heyr-i kalla-r heyr-ir kalla-r heyr-ir
Uni. Iceland - HUMANITIES - ÍSE102G
14. tmi Sasti tmi Srhljavxl stofni: B-vxl Notkun ntar dag Framt t sterkra sagnaTjning framtar Engin srstk sagnmynd til a tkna framt (kominn tma). Sagnmyndin nt er notu til a tkna framt Hann fer anga morgun Hn hringir brum aftur au flytja nsta mnui
Uni. Iceland - HUMANITIES - ÍSE102G
15. tmi Sasti tmi Framt t sterkra sagna Myndun Endingar dag Meira um t C-vxl Flokkun sterkra sagna Afturbeygt fornafnC-vxl Srhljavxl t (og lh.t. (past participle) sterkra sagna Kennimyndir sterkra sagna1 2 3 4 nh. t.et. t.ft. lh.t. brjta braut b
Uni. Iceland - HUMANITIES - ÍSE102G
16. tmi Sasti tmi C-vxl Flokkun sterkra sagna Afturbeygt fornafn dag Afturbeygt eignarfornafn BohtturAfturbeyging og eign eignarsambndum eru efn. notu til a tkna eiganda g skoa blai mitt selur blinn inn minn ef eigandi vsar til 1.p.et. inn ef eiga
Uni. Iceland - HUMANITIES - ÍSE102G
17. tmi Sasti tmi Afturbeygt eignarfornafn Bohttur dag t veikra sagnaVeikar og sterkar sagnir Flokkun sem byggist myndun tar Sterkar sagnir t me srhljavxlum (C-vxl) brjta, braut, brutum, broti Veikar sagnir t me viskeyti milli stofns og endingar
Uni. Iceland - HUMANITIES - ÍSE102G
18. tmi Sasti tmi t veikra sagna dag reglulegar veikar sagnir Notkun nokkurra httarsagna (modal verbs)reglulegar veikar sagnir Nokkrar algengar veikar sagnir hafa reglulega beygingu A) Nt eins og einn flokkur veikra sagna, t eins og annar flokkur v
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19. tmi Sasti tmi: reglulegar veikar sagnir Httarsagnir dag: Staaratviksor Samandregnar myndir spurnarsetningum OrarStaaratviksor Atviksor (ao.) beygjast ekki. Staaratviksor tkna: dvl sta (rest at a place), hreyfingu til staar (movement to a place)
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20. tmi Sasti tmi: Staaratviksor Orar dag: Forsetningar AukafallsliirFst fallstring Margar forsetningar (fs.) hafa fasta fallstringu, stra alltaf sama fallinu. f. - um, gegnum, kringum, . eir tala um myndina gf. a, af, fr, hj, nlgt, r, . Stelpan
Uni. Iceland - HUMANITIES - ÍSE102G
21. tmi Sasti tmi: Forsetningar Aukafallsliir dag: Spurnarsetningar SpurnarfornfnSpurningar A) j/nei spurningar Kemur hann dag? J, hann kemur dag Nei, hann kemur ekki dag B) spurningar me spurnarori Spurnaratviksor (beygjast ekki) t.d. hvenr, h
Uni. Iceland - HUMANITIES - ÍSE102G
22. tmi Sasti tmi: Spurnarsetningar Spurnarfornfn dag: Spurnarfornfn Spurnaratviksor Um prfihver + nafnor ef.ft. Til a spyrja um einn r hpi (partitive) g veit a a er bara einn strkur bekknum sem er fr Blgaru. Hver strkanna er a? hver er eintlu hve
Uni. Iceland - HUMANITIES - ÍSE202G
1.tmi Inngangur UpprifjunfrhaustmisseriNmstlun NmskeiibyggtuppeinsogSE102G MlfriIhaustmisseri. Beygingar. Njarnafnorabeygingar. kveinfornfn. Sagnbeyging: Setningalegatrii. vitengingarhttur,olmynd.Upprifjun Upprifjunhelstubeygingum. Upprifjunnokkru
Uni. Iceland - HUMANITIES - ÍSE202G
2.tmi Sastitmi dag Inngangur Upprifjunfrhaustmisseri. BeygingnafnoraII: KarlkynKk:gestur Et.nf.gestur f.gest gf.gesti ef.gests Ft.nf.gestir f.gesti gf.gestum ef.gesta hestur hest hesti hests hestar hesta hestum hestaKarlkynsor Algeng karlkynsnafnor
Uni. Iceland - HUMANITIES - ÍSE202G
3.tmi Sastitmi: dag: Njarbeygingarkk.:kkII Beygingnafnora Meiraumkarlkyn:kk.IIogkk.III Kvenkyn:kvk.IISrhljavxlstofni 1) Bvxl:,oy 2)Flknarihljavxl: a)ea a&gt;=Avxl a&gt;e=Bvxl Klofning(breaking)i&gt;j,i&gt;ja b)jija Alltafigf.et.ogaref.et.Hljavxl:jija nf. f
Uni. Iceland - HUMANITIES - ÍSE202G
4.tmi Sastitmi: dag: Njarbeygingarkk.ogkvk. Beygingnafnora: Stigbreytinglsingarora. Meiraumkvenkyn:kvk.III HvorugkynKvk.IIIkindskei mynd Et.nf.kind skei mynd f.kind skei mynd gf.kind skeiar myndar ef.kindar myndir Ft.nf.kindur skeiar skeiar myndir f
Uni. Iceland - HUMANITIES - ÍSE202G
5.tmi Sastitmi: dag: Njarbeygingarkvk.oghvk. Aeinsumstigbreytingulsingarora. framumstigbreytingulsingarora.Stigbreyting Srstkmyndaflo.ernotutilabera saman. Stofnlo.+viskeyti+endingar Viskeyti: Mst:(a)r Est:(a)st Mistigogefstastig Srstakarendingarf
Uni. Iceland - HUMANITIES - ÍSE202G
6.tmi Sastitmi: dag: Stigbreyting. Notkungreinis. bendingarfornfn.Greinir Nafnor geta veri kvein (indefinite) ea kvein(definite). kvein no. hafa viskeyttan greini inn. Greinirinn er settur aftan vi beygingarendinguno.:hesturinn. kveinno.hafaengangrein
Uni. Iceland - HUMANITIES - ÍSE202G
7.tmi Sastitmi: dag: Notkungreinis bendingarfornfn(fn.) Notkunveikrarbeygingarlo.bendingarfornafnis kk. Et.nf. s f. ann gf.eim eirri ef. ess Ft.nf. eir r f. gf. eim ef. eirra kvk. hvk. s a a v eirrar ess au r au eim eim eirraeirrabendingarfornafniess
Uni. Iceland - HUMANITIES - ÍSE202G
8.tmi Sastitmi: dag: bendingarfornfn Notkunveikrarbeygingarlsingarora Meiraumnotkunveikrarbeygingarlsingarora Eintluogfleirtlunafnor persnulegarsagnirNotkunveikrarbeygingarlo. 1) egar lo. stendur me no. me greini: Rauibllinnerbilaur Annahittigmlukonu
Uni. Iceland - HUMANITIES - ÍSE202G
9.tmi Sastitmi: Notkunveikrarbeygingarlsingarora Eintluogfleirtlunafnor persnulegarsagnir Aukafallsliir dag:Persnulegarsagnir Frumlagstendurnf. Beygingasamrmi:Sagnirlagasigafrumlagi tluog persnu. Andlgstandaaukafalli(f.,gf,eaef.): Beygingasamrmi:Lo.
Uni. Iceland - HUMANITIES - ÍSE202G
10.tmi Sastitmi: dag: persnulegarsagnir Aukafallsliir Meiraumaukafallslii Notkuna NotkunsjlfurAukafallsliir Falloraukafallinfallvalds,.e.n sagnareaforsetningarsemstrir aukafallinu. Aukafallsliirmerkjatma,staea magn(quantity,number,measurement): tmali
Uni. Iceland - HUMANITIES - ÍSE202G
11.tmi Sastitmi: Notkuna Notkunsjlfur dag: Tilvsunarsetningar kveinfornfn: allur enginnTilvsunarsetningar Byrjasamtengingunni(st.)sem. Komaoftastbeinteftirno.semreigavi. No.geturverimeeangreiniseftirvhvorta vsareitthvakveieakvei: Jnhittimanninn[sem
Uni. Iceland - HUMANITIES - ÍSE202G
12.tmi Sastitmi: allur enginn Tilvsunarsetningar kveinfornfn dag: kveinfornfn ekkineinn einhver nokkurekkineinn,ekkinokkur Merkingersvipuogorsinsenginn Pllenganbl =Pllekkineinnbl =Pllekkinokkurnbl. ekkineinnmeiranotaenekkinokkur neinnbeygisteins
Uni. Iceland - HUMANITIES - ÍSE202G
SE201G: Mlfri II 1. mars 201113. tmi KVEIN FORNFN bir, sumirINNGANGUR Annar, feinir, enginn, neinn, mis, bir, srhver, hvorugur, sumir, hver og einn, hvor og nokkur, einhver. Sj lka: allur, hvor tveggjaFORNFN UM TVO/ FORNFN UM RJ + HEILD &gt;2: allirH
Uni. Iceland - HUMANITIES - ÍSE202G
SE201G: Mlfri II 3. mars 201114. tmi KVEIN FORNFN hvorugur, annarFORNFN UM TVO/ FORNFN UM RJ + HEILD &gt;2: allirHLUTIsumir/nokkrir einn/hinn annar hver enginnNEITUN 2: bir 2annar/hinn annar hvor 1-1 1: einn/einhverhvorugur 1-1 enginnhvorugur, bls