LECTURE 6 TRANSFORMATION OF RANDOM VARIABLES

Transcription

1 LECTURE 6 TRANSFORMATION OF RANDOM VARIABLES

2 TRANSFORMATION OF FUNCTION OF A RANDOM VARIABLE UNIVARIATE TRANSFORMATIONS

3 TRANSFORMATION OF RANDOM VARIABLES If s a rv wth cdf F th Y=g s also a rv. If w wrt =g th fcto g dfs a mappg from th orgal sampl spac of S to a w sampl spac th sampl spac of th rv Y. g: S

4 -TO- TRANSFORMATION OF RANDOM VARIABLES Lt =g df a -to- trasformato. That s th qato =g ca b solvd ql: g E: Y=- =Y+ -to- E: Y=² =± sqrty ot -to- Wh trasformato s ot -to- fd dsjot parttos of S for whch trasformato s -to-.

5 If s a dscrt r.v. th S s cotabl. Th sampl spac for Y=g s ={:=g S} also cotabl. Th pmf for Y s fy P Y P f g g

6 Eampl Lt ~GEOp. That s Fd th p.m.f. of Y=- Solto: =Y+ f Y f f p p p p for 3... for 0... P.m.f. of th mbr of falrs bfor th frst sccss Rcall: ~GEOp s th p.m.f. of mbr of Broll trals rqrd to gt th frst sccss

7 Eampl Sppos ~Posso wth p! 0... Lt Y=4 =Y/4 /4 Th p / / 4!

8 CONTINUOUS RANDOM VARIABLE Lt b a rv of th cotos tp wth pdf f. Lt =g b dffrtabl for all ad o-zro. Th Y=g s also a rv of th cotos tp wth pdf gv b h f 0 g d d o. w. g for

9 Eampl Lt hav th dst f 0 0 othrws Lt Y=. =g =ly d=/d. h h. 0 log 0 othrws

10 Eampl Sppos has a potal dstrbto wth / f 0 0 Lt Y=4- =Y+/4 =g = Y+/4 d=/4d. / 4 0 h 4 0 ow..

11 TRANSFORMATION THAT ARE NOT -TO- Lt =g f th qato =g ca ot b solvd ql th ot o to o trasformato shold b sd. Wh trasformato s ot -to- fd dsjot parttos of S for whch trasformato s -to- ad s -to- trasformato or s cmlatv mthod.

12 Eampl Lt b a rv wth pmf 4 p 0 3 Lt Y= S ={ 0} ={04} 4 p p p p 3 7 p p p 3

13 Lt b a rv wth pmf p Eampl /5 / 6 /5 0 /5 / 30 Lt Y=. S ={ 0} ={04} /5 0 p 7 /30 7/30 4

14 Eampl Stors locatd o a lar ct wth dst f= othrws Corr crs a cost of U=6 wh sh dlvrs to a stor locatd at hr offc s locatd at / 4 4 d df f d P U F U U U U U

15 Eampl Lt hav th dst / f. Lt W=. Fd th pdf of W.

16 Frst stp Fd th dstrbto fcto of W Gw = P[W w] = P[ w] P w w f w 0 w w d F w F w F f whr

17 Scod stp Fd th dst fcto of W gw = G'w. d w F w F w dw f w w f w w d dw w w w w w w w f w 0.

18 TRANSFORMATION OF FUNCTION OF TWO OR MORE RANDOM VARIABLES BIVARIATE TRANSFORMATIONS

19 DISCRETE CASE Lt ad b a bvarat radom vctor wth a kow probablt dstrbto fcto. Cosdr a w bvarat radom vctor U V dfd b U=g ad V=g whr g ad g ar som fctos of ad.

20 Th th jot pmf of UV s V A U V U f v V U v f Pr

21 EAMPLE Lt ad b dpdt Posso dstrbto radom varabls wth paramtrs ad wth jot pmf.!! p Fd th dstrbto of Y = +. W d to df w varabl Y =. Th Y =0 Y =0 Y ad =Y ad =Y -Y.!! p ! p 0 0!!!! p

22 CONTINUOUS CASE Lt = hav a cotos jot dstrbto for whch ts jot pdf s f ad cosdr th jot pdf of w radom varabls Y Y Y k dfd as * g Y g Y g Y k k

23 If th trasformato T s o-to-o ad oto th thr s o problm of dtrmg th vrs trasformato ad w ca vrt th qato * ad obta w qatos as ** g g g k k k Assmg that th partal drvatvs st at vr pot k=. / g

24 Udr ths assmptos w hav th followg dtrmat J calld as th Jacoba of th trasformato spcfd b **. Th th jot pdf of Y Y Y k ca b obtad b sg th chag of varabl tchq of mltpl varabls. g g g g dt J

25 As a rslt th w p.d.f. s dfd as follows: othrws J g g g f g 0 for

26 Lt ad ~ Ep Eampl f 0 0 Cosdr th radom varabls Y= Y=+. Fd pdf of Y ad Y? =Y ad =Y-Y th Jacoba s 0 Th f f 0

27 METHOD OF CONDITIONING U=h Fd f b trasformatos Fg = Obta th jot dst of U : f = f f Obta th margal dstrbto of U b tgratg jot dst ovr f U f f d

28 Eampl ~Btaab ~Btaa3b Idpdt U= F = ad gt f 0 l 8 l l / / 6 0 / / 6 / / d d f f f f f f f du d U U f f U

29 Eampl dpdt Epotalq f =q - -/q >0 q>0 = f = q - -+/q >0 U= + ~ 0 / / / / / / / 0 / 0 / / 0 0 / / 0 / / 0 0 q b a q q q q q q q q q q q q q q q q q q q q q q q q Gamma U f d d d d d d P U U U U

30 M.G.F. Mthod If ar dpdt radom varabls wth MGFs M t th th MGF of s Y M t M t...m t Y

31 Eampl dpdt Lt ~ B p Th fd th pmf of M t M t... M t Y k Y k t t k p q... p q t... k p q k ~ B p. M t M t... M t Y k t t k p q... p q t p q... k k

32 Eampl b a b b b b b a a a a a ~... dpdt... ~... t t t ty Y Gamma Y t t t t M t M E E E t M Y t t M Gamma

33 Eampl ta ta a a t ty Y a a Normal a Y t a t a t a t a a t a t t a M a t M E E E t M a a Y t t t M Normal... ~ p p p fd costats } {... p dpdt... ~

34 ORDER STATISTICS

35 ORDER STATISTICS Lt b a r.s. of sz from a dstrbto of cotos tp havg pdf f a<<b. Lt b th smallst of b th scod smallst of ad b th largst of. a b s th -th ordr statstc. m ma

36 ORDER STATISTICS It s oft sfl to cosdr ordrd radom sampl. Eampl: sppos a r.s. of fv lght blbs s tstd ad th falr tms ar obsrvd as Ths wll actall b obsrvd th ordr of Itrst mght b o th kth smallst ordrd obsrvato.g. stop th prmt aftr kth falr. W mght also b trstd jot dstrbtos of two or mor ordr statstcs or fctos of thm.

37 JOINT PDF OF THE OREDER STATISTICS If s a r.s. of sz from a poplato wth cotos pdf f th th jot pdf of th ordr statstcs s! g f f f for... Ordr statstcs ar ot dpdt. Th jot pdf of ordrd sampl s ot sam as th jot pdf of ordrd sampl. Not: For dscrt dstrbtos w d to tak ts to accot two s bg qal.

38 Eampl Fd th jot pdf of th ordr statstcs for th form dstrbto th stadard potal dstrbto ad ormal dstrbto? Solto: p.d.f for th form s: f 0 g...! 0...

39 Solto: p.d.f for th stadard Epotal dstrbto s: f 0...! 0... g

40 Solto: p.d.f for th stadard ormal dstrbto s: f! g......

41 Eampl Sppos that ad 3 rprst a radom sampl of sz 3 from poplato wth pdf f 0 Jot pdf of ordr statstcs Y Y ad Y 3? g 3! Margal pdf of Y 3 3 g 48 d d 6 0

42 THE MARGINAL DISTRIBUTIONS FOR THE ODER STATISTICS Thorm: p.d.f of th rth ordr statstcs If b a r.s. of sz from a poplato wth cotos pdf f th th p.d.f. of th rth ordr statstcs r s gv as:! r gr r f r [ F r ] [ F r ] r r! r!

43 r-th Ordr Statstc r- r r+ P< r P> r f r! g r r f r r! r! # of possbl ordrgs!/{r!! r!} r [ F ] r [ F r ] r

44 Eampl ~Uform0. A r.s. of sz s tak. Fd th p.d.f. of kth ordr statstc. Solto: Lt Y k b th kth ordr statstc. g Y k k!! k! k k k k k k for 0 Y k ~ Btak k

45 Eampl Sppos that rprst a radom sampl of sz from poplato wth pdf f 0 Margal pdf of Y ad Y? F 0 g 0 g 0

46 Thorm p.d.f of th largst ordr statstcs: If b a r.s. of sz from a poplato wth cotos pdf f th th p.d.f. of th Largst ordr statstcs Y s gv as: g f F [ ]

47 Thorm: p.d.f of th smallst ordr statstcs If b a r.s. of sz from a poplato wth cotos pdf f th th p.d.f. of th smallst ordr statstcs s gv as: g f [ F ]

48 Eampl Lt Y Y... Y 6 ar a O. S. of sampl sz = 6 ad th p.d.f. of ths sampl s Fd: f 0 g g g r r 6 6 Solto: 6! g r!6 r! r 6 r r [ ] 0 6 r r r 6 5 g [ 6 ] g

49 JOINT P.D.F. OF -TH AND j-th ORDER Thorm: STATISTIC FOR <j If b a r.s. of sz from a poplato wth cotos pdf f ad Y < Y < <Y ar th ordr statstcs of that sampl th th p.d.f. of th two ordr statstcs Y < Y j <j ad j = s gv as! g F f F F f F! j! j! j j j j [ ] [ j ] j [ j ]

50 Jot p.d.f. of -th ad j-th Ordr Statstc for <j - tms j-- tms -j tms tm tm - + j- j j+ # of possbl ordrgs!/{!!j--!! j!} P< P <<j P> j f f j! g F f F F f F! j! j! j j j j [ ] [ j ] j [ j ]

51 Eampl Sppos that rprst a radom sampl of sz from poplato wth pdf f 0 Fd th dst of rag R=Y -Y? F 0! g 0!