Contents. Random Variables

Transcription

1 Rado Sgal Processg Chapter Rado Varables Chapter Rado Varables Cotets Rado Varables.... Deto o a Rado Varable..... Cuulatve Dstrbuto Fucto CDF..... Probablt Dest Fucto PDF..... partal characteratos Codtoal Cuulatve Dstrbuto Fucto Characterstc ucto Hgher-Order Moets or Gaussa Rado Varable.... Coo Cotuous Rado Varable.... Coo Dscrete Rado Varables...6. Trasoratos o Oe Rado Varable Trasorato o Oe Rado Varable Cuulatve Dstrbuto Fucto....5 Coputato o pected Values....6 Two Rado Varables Jot Cuulatve Dstrbuto Fucto Jot Probablt Dest Fucto Partal Characteratos Jotl Noral Rado Varables....7 Two Fucto o Two Rado Varables Probablt Dest Fucto Dscrete Rado Varables Probablt Dest Fucto Cotuous Rado Varables ad Cotuous Fuctos.7. Dstrbuto Fucto Cotuous Dscrete or Med....8 Oe Fucto o Two Rado Varables Probablt Dest Fucto Dscrete Rado Varables Probablt Dest Fucto Cotuous Rado Varables....9 Coputato o h.... Multple Rado Varables..... Total Characteratos..... Partal Characteratos..... Gaussa Rado Vectors.... M Fuctos o N Rado Varables.... Suar... da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

2 Rado Sgal Processg Chapter Rado Varables Rado Varables. Deto o a Rado Varable A eperet s speced b the three tuple S F P where S s a te coutable or ocoutable set called the saple space F s a Borel eld specg a set o evets ad s a probablt easure allowg calculato o probabltes o all evets. Usg a uderlg eperet a rado varable S that satses the ollowg: a e P e s deed as a real-valued ucto o e : s a eber o F or all whch guaratees the estece o the cuulatve dstrbuto ucto ad b the probabltes o the evets e : e ad e + or te wth a ero probablt. e : are both ero. Ths eas that the ucto s ot allowed to be aple. s speced b F P where S a b c d F s the power set o S ad S deed b P a. P b. P c. ad P d. P s. Dee the ollowg appg a b c d. Is the ucto. a rado varable? Soluto To show that the ucto s a rado varable t ust be show that codtos a ad b speced above are satsed. The sets e : e as vares ro to are as ollows: For e : e e : e b e : e a b c e : e a b c d S Sce the sets descrbed are subsets o S the are ebers o the power o S ad thus codto a s satsed. I S has a te uber o eleets ad F s the power set o S codto a wll alwas be satsed. I the power set s ot used or F t s possble to costruct a ucto that s ot a rado varable. It s easl see that P P e : e P e : e P Thus codtos b s satsed ad sce a s also satsed. s a rado varable. Most coo uctos deed o a gve eperet are rado varables however codto a da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

3 Rado Sgal Processg Chapter Rado Varables ca be easl volated F s ot the power set o S or codto b caot be satsed s deed to be or a set wth te probablt. Rado varables are sad to be totall charactered or descrbed wth relato to calculatg probabltes o acceptable evets.e. evets that are a eber o F b ther cuulatve dstrbuto ucto or probablt dest ucto. Weaer characteratos called partal characteratos would clude specg hgher-order oets varace ea ad the le. Kowg ust the ea o a rado varable s certal less orato about the rado varable tha owg the probablt dest ucto et t stll provdes soe dea about values o the rado varable... Cuulatve Dstrbuto Fucto CDF The cuulatve dstrbuto ucto represeted b ust s deed or all as F F or a rado varable e or whe coveet P{ e : e } P{ }. It s sucet orato to calculate the probabltes o all allowable evets ad as a result s called a total characterato. Because o the propertes o the probablt easure P descrbed the overall eperet the cuulatve dstrbuto ucto CDF ca be show to have a uber o portat propertes: F s bouded ro above ad below F s a odecreasg ucto o F or all. F F or all ad all. F s cotuous ro the rght l F a a F F ca be used to calculate probabltes o evets P{ P{ P{ P{ } F } F } F } F F F F F Where l F let-had lt 5 Relato to the probablt dest ucto to be deed later s wrtte as..5 da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

4 Rado Sgal Processg Chapter Rado Varables F d.6 aple. For the rado varable deed aple. detere a the cuulatve dstrbuto ucto F ad calculate the probabltes o the ollowg evets. Usg ths dstrbuto ucto detere b P c P d P P P g P h P P ad P. Soluto a The CDF F or the rado varable s obtaed b deterg the probabltes o the evets e : e or all. The results ro aple. help us detere F as ollows or the ollowg regos. e : e P e : e Pb. e : e Pa b c Pa Pb Pc e : e Pa b c d P S.9 These results ca be plotted as show. The CDF alog wth the probabltes o the tervals gve q..5 wll be used to detere the probabltes o the evets lsted. a P F F.9.9 b P F F.9. c P.9. 9 F F d P.. F F e P.. F F P F F.9. g P F. h P F F P.9.9 F F e.. Probablt Dest Fucto PDF The probablt dest ucto or a rado s a total characterato ad s deed da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

5 Rado Sgal Processg Chapter Rado Varables as the dervatve o the cuulatve dstrbuto ucto d F.7 d I F has up dscotutes t s coveet to use delta uctos so that a probablt dest ucto PDF wll alwas be deed. Thereore a probablt dest ucto wll cota delta uctos at the pots o dscotutes o F wth weghts equal to the se o the up at those pots. Iportat propertes o the probablt dest ucto or a rado varable are as ollows: postvt Itegral over all ut area or all.8 used to calculate probablt o evets d.9 p{ } d d p{ p{ } } d d d d. p{ } d d Relatoshp to cuulatve dstrbuto ucto df. d A rado varable s called a dscrete rado varable ts probablt dest ucto s a su o delta ucto ol or correspodgl ts cuulatve dstrbuto ucto F s a starcase ucto. A rado varable s called a cotuous rado varable ts cuulatve dstrbuto ucto has o te dscotutes or equvaletl ts probablt dest ucto has o delta uctos. I a rado s ether a cotuous or dscrete rado varable we call t a ed rado varable. da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 5 o 58

6 Rado Sgal Processg Chapter Rado Varables Dscrete rado varable Rado varable Cotuous rado varable Med rado varable aples o these three tpes ollow. Dscrete rado varable Cotuous rado varable Med rado varable.... e....5e F F F a Dscrete RV b Cotuous RV c Med RV Fgure. aples o a dscrete b cotuous ad c ed rado varables. aple.5 da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 6 o 58

7 Rado Sgal Processg Chapter Rado Varables Gve a ed rado varable wth probablt dest ucto Detere the cuulatve dstrbuto ucto Soluto The cuulatve dstrbuto ucto F 8 e d e e F. F ca be detered b the relatoshp gve.6. d The tegrals o the delta ucto as goes through the values gves ut stepscausg dscotutes or ups at the pots o ses 8 ad 8 respectvel. d.. partal characteratos Iportat partal characteratos o rado varables are the ea varace hgher-order oets ad cetral oets ad codtoal dstrbuto ad dest uctos. Detos or these partal characteratos ollow. The ea varable s deed b o a rado varable or equvaletl the epected value o a rado d. The ea o a rado varable s also reerred to as the average o the rado varable as see ro the tegral above. Kowg the ea o a rado varable s ot sucet orato to calculate probabltes o gve evets. It s ust barel eough orato to boud probabltes o soe specal evets as see the ollowg equalt or the specal case whe the rado varable taes o postve values: P{ a} or a. a The varace o a rado varable s deed b d. The stadard devato s deed as the square root o the varace ad deoted as.the varace da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 7 o 58

8 Rado Sgal Processg Chapter Rado Varables da Uverst Lu Cogeg Page 8 o 58 gves a sese o spread o the rado varable as larger values would dcate a wder dest ucto ad as wth the ea t s ot sucet orato or calculato o probabltes o evets. However le the ea bouds ca be placed o the calculato o soe specal evets. Two o the ost portat bouds are descrbed the Tchebche equaltes or the evets { } ad } {. Notce that these evets represet the regos correspodg to the ea stadard devatos about the ea ad the copleet o that rego. The Tchebche equaltes correspodg to bouds o the probabltes o the tals ad core o the dest ucto respectvel or a dest ucto ad are } { } { P P.5 For a gve postve we ca also boud the } { P whch s the probablt that s the rego aroud the ea b } { P.6 q..5 ca be obtaed ro. b lettg. } { : Proo d d d P } { } { equaltes : Tchebche P or P The hgher-order oets ad hgher-order cetral oets o a rado varable are partal characteratos ad provde addtoal orato about the statstcs propertes o rado varables. The are deed as ollow or : d.7 d.8 A geeralato o the Tchebche equalt called the equalt o Beae gves a boud o the evet } { a hgher-order oets about a arbtrar value a ot ecessarl the ea as

9 Rado Sgal Processg Chapter Rado Varables I a ad a P{ a }.9 the equalt above reduces to Tchebche equalt... Codtoal Cuulatve Dstrbuto Fucto Aother partal characterato o a rado varable s the codtoal dstrbuto ucto deed or codtoal evet C b F C P{ C}. The deterato o F C or a gve evet C ad a gve F or a rado varable uses the deto o codtoal probablt as ollow: F P{ C} C. P C Slarl the codtoal probablt dest ucto or a rado varable codtoed o the evet C s deed the ters o the dervatve o the codtoal dstrbuto ucto as df C C. d I C s gve ters o eleets o the saple space S the the P{ C} ca be detered b worg wth the orgal saple space. I a probles the codtog evet s gve ters o the values o the rado varable. aple.6 A rado varable s charactered b ts cuulatve dstrbuto ucto F e ad a evet C s deed b C.5. Detere the codtoal cuulatve dstrbuto ucto F C ad codtoal probablt dest ucto C or the rado varable codtoed oc. Soluto B q.. the codtoal dstrbuto ca be wrtte as F C P{.5 } P.5 The uerator wll be deret depedg o where s the te terval as da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 9 o 58

10 Rado Sgal Processg Chapter Rado Varables or or or.5 But we ow that the P P C P{.5 } P.5 e e e e e e e. 865 P{.5 } P.5 F e P{.5 } P.5 F s detered as C P{.5 } P.5 F.5 e e. 865 F F C.5 C e e F C F F.5 Dvdg the probabltes b PC gves the al aswer or the codtoal dstrbuto as or.5 or.5 F.865 or Plots o the F ad C F are show or ths eaple Fgure The codtoal probablt dest ucto ca be obtaed b tag the dervatve o the codtoal dstrbuto throughout the rego o the codtog evet ad the result s C.957e.5 elsewhere F C F.5.5 Fgure.6 F ad F C or aple.6..5 Characterstc ucto The characterato o a rado varable ca also be gve b specg ts characterstc ucto whch s the oded Fourer trasor o ad deed as da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

11 Rado Sgal Processg Chapter Rado Varables e d e F. e d Speccato o characterstc ucto s equvalet to a total characterato sce ts verse trasor elds the probablt dest ucto. aple.7 Detere the characterstc ucto or a Gaussa rado varable wth ow ea ad varace. Soluto B the deto above the characterstc ucto becoes ep e d Ater epadg out the ter the bracets ad cobg t wth the other epoet over a coo deoator the characterstc ucto becoes ep d Net coplete the square the bracets ad ultpl b the proper correcto so that the characterstc ucto becoes ep ep d The epoetal ter o the rght ca be tae through the tegral sg sce t s ot a ucto o ad sce the tegral s o a Gaussa dest t gves a value o. We ca all wrte ater splg the secod epoet the characterstc ucto or a Gaussa rado varable as ep. The hgher-order oets ca also be detered ro the characterstc ucto as.5 The above equato ca be used to detere the oets o a rado varable b obtag the characterstc ucto or oded Fourer trasor ad perorg the deretato ad evaluato at ero. da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

12 Rado Sgal Processg Chapter Rado Varables da Uverst Lu Cogeg Page o 58 aple.8 A rado varable has a probablt dest ucto gve b ae a. Detere the oets b usg the characterstc ucto. Soluto The characterstc ucto ca be obtaed b tag the oded Fourer trasor o the as a a a a ae F d e a B q..5 the oets are detered as ollows: a a a a a a a a a a O cotug ths process t s eas to see that a a a a a Aother ucto that ca be used to detere the oets or a rado varable s the oet-geeratg ucto deed as ollows: t t e d e t M.6 As the sae ples the oet-geeratg ucto ca be used to calculate the oets or a rado varable or whch the dervatves est ad relatoshp ca be show to be t t t M.7 The oet-geeratg ucto does ot alwas est but whe t does t provdes a alteratve to usg the characterstc ucto.

13 Rado Sgal Processg Chapter Rado Varables da Uverst Lu Cogeg Page o Hgher-Order Moets or Gaussa Rado Varable Let be a Gaussa rado varable wth ow ea ad varace. The hgher-order oets ca be wrtte ters o the probablt dest ucto as d d } ep{.8 I s replaced b the tegral ca be wrtte as d } ep{.9 Ater epedg the the hgher-order oets are wrtte as d } ep{. But deed b s a Gaussa rado varable wth ero ea ad varace equal to the varace o. B setr o the ero ea Gaussa dest all hgher-order odd oets o are ero ad. ca be wrtte as odd eve. The hgher-order oets up to order s are easl detered ro. as ollows: I the t s see ro the above that or as epected.. Coo Cotuous Rado Varable Gaussa Rado Varable. A rado varable s called a Gaussa or oral rado t has a

14 Rado Sgal Processg Chapter Rado Varables probablt dest ucto gve b ep g ;. It ca be see that ad are the ea ad varace o the Gaussa rado varable. The stadard devato s deed as. Rather tha wrte the epresso. t s coo practce to use the otato ~ N Whch eas that s a Gaussa rado varable wth ea ad varace. ollows: The hgher-order cetral oets ca be detered or the Gaussa dest ters o the as or eve. or odd Uor Rado Varable. A rado s uor ts probablt dest o a terval s gve b a b b a.5 otherwse The ea ad varace o a uorl dstrbuted rado varable ca be show to be a b b a.6 poetal Rado Varable. A epoetal rado varable has the ollowg probablt dest ucto or a : ae a otherwse The ea ad varace or a epoetal dstrbuted rado varable are.7.8 a a Ralegh Rado Varable. A Ralegh dstrbuted rado varable has or b the ollowg probablt dest ucto: ep otherwse.9 da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

15 Rado Sgal Processg Chapter Rado Varables The ea ad varace or a Ralegh dstrbuted rado varable are /. Gaa Rado Varable. A Gaa dstrbuted rado has or ad the ollowg probablt dest ucto: Where e a / e otherwse I s a postve teger the t s well ow that!. The ea ad varace or a Gaa dstrbuted rado varable are d.. Cauch Rado Varable. A rado s Cauch dstrbuted rado ts probablt dest taes the ollowg or or a : a. a The ea ad varace or a Cauch dstrbuted rado varable are does ot est does ot est. Ch square Rado Varable. A rado varable s ch-squared dstrbuted wth degree o reedo N N / ep.5 N / N / The ea ad varace or a Ch-square rado varable are N N.6 Log Noral Rado Varable. A rado varable s log orall dstrbuted ts probablt dest ucto s o the or ep log otherwse e.7 da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 5 o 58

16 Rado Sgal Processg Chapter Rado Varables The ea ad varace ters o the postve paraeters ad are ep / ep ep.8 Beta Rado Varable. A Beta-dstrbuted rado varable has or ad the probablt dest ucto I ad!. otherwse ad e d are postve tegers the!! The ea ad varace or a beta-dstrbuted rado varable ca be show to be.9 ad.5. Coo Dscrete Rado Varables Beroull Rado Varable. A dscrete rado varable s Beroull ts probablt dest s gve b p p.5 The ea ad varace or a Beroull dstrbuted rado varable ca be show to be p p p.5 Dscrete Uor Rado Varable. A dscrete rado varable s uor ts probablt dest o a rage s gve b b a a.5 The ea ad varace or a uorl dstrbuted rado varable ca be show to be a b b a.5 Posso Dstrbuto. A rado varable s Posso dstrbuted ts probablt dest ucto s gve b e!.55 da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 6 o 58

17 Rado Sgal Processg Chapter Rado Varables The ea ad varace or a Posso dstrbuted rado varable ca be show to be.56 Boal Rado Varable. A rado varable s boall dstrbuted ts probablt dest ucto s gve b p p.57 The ea ad varace or a boall dstrbuted rado varable ca be show to be p p p.58 could be terpreted as the uber o successes repeated trals o the Beroull tpe. Geoetrc Rado Varable. A rado varable s geoetrcall dstrbuted ts probablt dest ucto s gve b p p.59 The ea ad varace or a geoetrc rado varable ca be show to be P.6 P P could be terpreted as the uber o the tral or whch the rst success occurs or repeated trals o the Beroull tpe. Negatve Boal Rado Varable. A rado varable s Negatve boall dstrbuted ts probablt dest ucto s gve b p p.6 The ea ad varace or a egatve boall dstrbuted rado varable ca be show to be.6 P P p could be terpreted as the uber o the tral o whch the th success occurs or repeated trals o the Beroull tpe.. Trasoratos o Oe Rado Varable Let e be a rado varable deed o a saple space S. Dee a real-valued ucto the real ubers. I e or all e a eber o S s the doa o the g g o the the rage da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 7 o 58

18 Rado Sgal Processg Chapter Rado Varables becoes the set o all e such that e g e. Ths ca be descrbed b e e g e e or all e eber o S.6 As we usuall drop the e ro the rado varable e ad use ust t s epedet to drop the de ro e ad use ust. Thereore q..6 s usuall represeted as g.6 g e Saple space e Reals ge Reals S g g g Fgure.7 Trasorato o rado varables as a appg ad a tabular.. Trasorato o Oe Rado Varable I geeral uless g s a lear ucto o descrbed b g a b wth a ad b costats or other specal orato regardg the tpe probablt ad tpe o ucto are speced the epected value o caot be detered. I geeral or g. o the ea that s g the ea o s ot the ucto g.65 Probablt Dest Fucto Dscrete Rado Varable. I s a real dscrete rado varable t taes o a te or coutable set S o possble values. da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 8 o 58

19 Rado Sgal Processg Chapter Rado Varables Thereore s totall charactered b ts probablt dest ucto cosstg o a su o weghted pulse ucto at the as ollows: S p.66 I the epresso above p s the probablt that or ore precsel s deoted b P{ e : e }. I taes o ol the values t s reasoable that taes ol o the values g. Thus s a dscrete rado varable whose probablt dest ucto ca be wrtte as S p g.67 I soe cases the olear ucto g s such that g o several deret gve the sae value Where thus allowg us to wrte the dest as p.68 S S s the set o all uque g ad p s the su o the probabltes. aple.9 Let be a dscrete rado varable charactered b ts probablt dest ucto as ollows: Dee the ucto g as show Fgure.8. =g 5 = Fgure.8 Fucto =g or aple.9 Soluto da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 9 o 58

20 Rado Sgal Processg Chapter Rado Varables ach pot s tae through the trasorato ad le values are collected to gve Probablt Dest Fucto Cotuous Rado Varable. Theore. I s a cotuous rado varable charactered b ts probablt dest ucto ad g s a cotuous ucto wth o lat spots te legths o costat value the the probablt dest ucto as ollows whch charactered the rado varable g Where S s gve or each value o.69 dg / d S s the set o all real solutos o g o real soluto o g ests The theore above ca be used to d the probablt dest ucto requres solvg g or each value o ro to. or g ad t aple. Gve a rado varable wth probablt dest ucto d the probablt dest ucto or a rado varable deed b as show Fgure.9. Wor the proble or the ollowg cases: a b ep.5ep. u. = > < Fgure.9 Trasorato = or aple. da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

21 Rado Sgal Processg Chapter Rado Varables Soluto a Codtos are satsed to use the udaetal theore drectl as g s a cotuous ucto ad s a cotuous rado varable. For a has o real soluto the dotted le or a arbtrar does t tersect the curve I the gves or. has two real roots. Thereore or those values o. ad. Usg the udaetal theore Usg the two-sded-epoetal dest or b Usg ep ca be spled to.5e.5e e t s see that or ad that or e e e I s a cotuous rado varable ad g s a cotuous ucto ecept or a te or coutable uber o lat spots the the probablt dest or g ca be oud b a slght odcato o the udaetal theore. The probablt dest ucto s coposed o two parts. For all values o such that there are o lat spots d the dest as gve b the udaetal theore ad the add to ths su o delta uctos oe or each lat spot at probablt that the rado varable produces tpcal cotuous ater passg through wth weght equal to the g. I Fgure. a g s show that cotas lat spots at or values o o the tervals respectvel. I ths otato the probablt dest ucto ca be wrtte as dg / d o real p soluto o g est.7 da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

22 Rado Sgal Processg Chapter Rado Varables = g Flat Flat Flat Fgure. Fucto = g that cotas lat spots aple. Let be a Gaussa rado varable wth ero ea ad ut varace. The ucto g s deed Fgure. ad the rado varable s speced b the trasorato g. Fd the probablt dest ucto or. Soluto I ths eaple the ucto g has two lat spots oes at 9 ad the other at. Fro the graph o probablt that g t s see that probablt that 9 s the probablt that equals the probablt that. paraeters gve these probabltes becoe where 9 P P ep whle the. Usg the Gaussa dest ad the P ep d. 8 P ep d. d. Treatg the cotuous part o the trasorato separatel t s see that there are our deret stuatos as goes ro to. For g has o real roots thereore. For 9 aga o real roots so. For there are two real roots at ad. Thereore c s gve b da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

23 Rado Sgal Processg Chapter Rado Varables c ep ep ep For 9 g has ol oe real root at ad b the udaetal theore c becoes ep c as Usg ut step uctos the cotuous ad dscrete parts ca be cobed to gve the al aswer ep.8. 9 ep 9 Probablt Dest Fucto Med Rado Varable. I s a ed cotuous varable ts probablt dest ucto ca be wrtte as where c c p.7 represets the dest wthout delta uctos. I g s a cotuous ucto the ca be obtaed b addg the results ro usg the udaetal theore o c to the results obtaed b hadlg the pulses as prevousl descrbed or trasorato o dscrete rado varable. I c g has lat spots the the odcato o the udaetal theore ca be used o wth the results added to the dscrete part... Cuulatve Dstrbuto Fucto Assue that a rado varable s charactered b ts probablt dest ucto ad that a rado varable s deed b g where g s a real-valued ucto. It s desred to d the cuulatve dstrbuto ucto F drectl ad the use t to obta the deto o a cuulatve dstrbuto ucto gves. Usg the basc F P{ } P{ g } P{ : g } d.7 I da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

24 Rado Sgal Processg Chapter Rado Varables where { : g }. Ths rego s llustrated Fgure.. I =g I Fgure. Rego o tegrato I to obta the cuulatve dstrbuto ucto. aple. Assue that s a rado varable wth probablt dest ucto gve b ep Dee the rado varable b g where g dstrbuto ucto F or the rado varable. Soluto s show Fgure.5. Fd the cuulatve To d F usg q..7 t s ecessar to det the regos I or all values o. or ths proble t turs out that there are our dstct regos or deret rages o as show Fgure.. Rego. For : g s the ull set whch results F. Rego. For g : I :. Thereore becoes F Rego. For 9 F ep I d : ad s F F ep d Rego.For 9 all I ; thereore F F d. Suarg the results or the our regos above the cuulatve dstrbuto ucto F becoes da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

25 Rado Sgal Processg Chapter Rado Varables F whch s llustrated Fgure >9 =g <<9 8 6 << < - I Fgure. Rego or aple -. F.8. Not to scale 9 Fgure. Cuulatve dstrbuto ucto or aple. da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 5 o 58

26 Rado Sgal Processg Chapter Rado Varables Whe tag the dervato o a tegral wth respect to a varable that appears the lts o the tegral t s actuall epedet to use the Lebt rule whch ts ost geeral or s d d d d d d d d d.7 d.5 Coputato o pected Values For a rado varable wth probablt dest ucto rado varable ca be deed b value o g. Method. B usg whch elds Method. B usg whch elds ad a real-valued ucto g g. There are three coo was to copute the epected d.7 g g d.75 Method. B usg the Mote Carlo techque whch essetall sthetc saplg. Method represets a alteratve to ethod ad ethod ad t s ot a aaltcal ethod. Mote Carlo Saplg s useul probles where the tegrals are o such a ature that aaltcal solutos are ot easl obtaed or do ot est. Roughl speag t s a eperetal ethod usg sthetc saplg ro the dest o to geerate saples coputg saples coputg the average o those results to get the approato or. a g ad the aple. I aple. a rado varable was deed as For ths proble copute the Soluto g ad was gve b ep usg ethods ad. Method. The dest detered aple.9 s used to copute the tegral da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 6 o 58

27 Rado Sgal Processg Chapter Rado Varables B ag a chage o varables e d d becoes e d e Method. the orgal dest or ad g s used: e g g d d e d As ths eaple shows the relatoshp betwee the two ethods s a chage o varables wth the tergrato. Clearl the probablt dest ucto or s ot eeded ts calculato s a uecessar step the process o coputg the epected value ad the secod ethod would be preerred..6 Two Rado Varables A eperet s speced b the three-tuple S F P where S s a te coutable or ocoutable set called the saple space F s a Borel eld specg a set o evets ad P s a probablt easure allowg calculato o probabltes o all evets. Based o ths uderlg eperet two rado varables e ad e are deed as real-valued uctos o the sae S that satses the ollowg codtos: a { e : e ad e } s a eber o F or all ad. Ths guaratees the estece o the cuulatve dstrbuto ucto. b The probabltes o the evets { e : e } { e : e } { e : e } ad { e : e } are all ero. Ths eas that the ucto s ot allowed to be + or t wth a oero probablt..6. Jot Cuulatve Dstrbuto Fucto The ot cuulatve dstrbuto ucto F or a rado varable e ad e represeted b ad s deed or all ad as da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 7 o 58

28 Rado Sgal Processg Chapter Rado Varables F P{ e : e e } P{ }.76 It s sucet orato to calculate the probabltes o all allowable evets ad thus s called a total characterato. Because o the propertes o the probablt easure P descrbed the overall eperet the ot cuulatve dstrbuto ucto ca be show to have a uber o propertes. F s bouded ro above ad below F s a odecreasg ucto o ad F or all ad.77 F F or all all ad all F F or all all ad all.78 F s cotuous ro the rght both ad l F F F ca be used to calculate probabltes o rectagular evets as P {.79 } F F.8 5 F s related to the ot probablt dest ucto b F dd.8 aple. Rado varables ad are deed Fgure.6. I ths eaple ad are both dscrete rado varables. Fd the ot cuulatve dstrbuto ucto F F ad F or ad. Soluto Fro the table t observed that taes o values ad wth te probabltes o.6 ad. respectvel whle taes o values ad wth probabltes o.. ad. respectvel. Thereore F ad are F F.6. F... To obta the ot dstrbuto t s ecessar to copute probablt o the evet as ad are both vared ro to. Ths wll be doe b g deret regos ad deterg the probablt o as vares da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 8 o 58

29 Rado Sgal Processg Chapter Rado Varables 5 6 P ad all the set For so F. For there wll be three deret regos o the =as or whch deret values o the dstrbuto are obtaed as F F P P P P P... P.... F For the results are slar to above but there are our deret regos that gve the ot cuulatve dstrbuto ucto as F Fgure.6 Rado Varables ad or aple - P F F F P P P P... P P P PS The ot cuulatve dstrbuto ucto s show Fgure Jot Probablt Dest Fucto The ot probablt dest ucto or the rado varable ad s a total characterato ad s deed as the dervatve o the ot cuulatve dstrbuto ucto. F.8 I F has up dscotutes t s coveet to use delta uctos so that a ot probablt dest ucto wll alwas be deed. Thereore a probablt dest ucto wll cota delta uctos at the pots o dscotutes o F wth weghts equal to the se o the ups at those pots. da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 9 o 58

30 Rado Sgal Processg Chapter Rado Varables CDF Fucto F Iportat propertes o the probablt dest ucto are as ollow: Postvt Itegral over all ad or all ad.8 ca be used to calculate probablt o rectagular evets as Where Fgure.7 Cuulatve dstrbuto ucto F or aple -.8 P } dd.85 { ad are lts ro the postve sdes or a evet A as Relatoshp to ot cuulatve dstrbuto ucto P { } A dd.86 A dd F.87 Rado varables are called otl dscrete rado varables ther probablt dest ucto s a su o two desoal delta uctos ol ad correspodgl ts cuulatve dstrbuto ucto F s a bo starcase tpe ucto. Rado varables are called otl cotuous rado varables ther cuulatve dstrbuto ucto has o te dscotutes or equvaletl ts probablt dest ucto has o delta uctos. I rado varables are da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

31 Rado Sgal Processg Chapter Rado Varables ether otl cotuous or otl dscrete rado varables the are otl ed rado varables. aple.5 The rado varables descrbed aple. are otl dscrete rado varables. Gve the ot probablt dest ucto or these rado varables ad. Soluto The ot dest s see to be Partal Characteratos Iportat partal characteratos or two rado varables are the argal destes eas varaces covaraces hgher-order ot oets ad ot cetral oets. For two rado varables ad the argal destes or ad are deed as the destes o ad b theselves ad wll be deoted b ad as beore. These argal destes ca be obtaed ro the ot probablt dest ucto or the two rado varaces as d d The codtoal probablt dest uctos or ad are deed as The eas ad or two rado varables are ad usg the codtoal destes dee the codtoal eas as d d.9 da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

32 Rado Sgal Processg Chapter Rado Varables Jot Dest Codtoal Dest = /A Prole A Area d Fgure.8 Codtoal probablt dedt uctos B rearragg q..89 we ca rewrte the ot probablt dest ucto ters o the codtoal ad argal destes as.9 B substtutg.9 to.88 alteratve orulas or deterg the argal destes or ad ca be oud as ollows: d d.9 ag.9 we see that t s also possble to wrte the relatoshps betwee the two codtoal destes as / /.9 These orulas are coparable to Baes s rule whch epresses the relatoshp betwee codtoal probabltes ecept that the represet the relatoshps betwee the codtoal probablt dest uctos ad ot probabltes. Should ether or be dscrete rado varables these results would eed to be re-wrtte. For eaple let be a dscrete rado varable ad a cotuous rado varable wth ow { } or all ad ow : da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

33 Rado Sgal Processg Chapter Rado Varables N P{ The t s easl show or all to N that }.9 { } P{ } P{ } N.95 { } P{ } aple.6 Gve the ot probablt dest ucto 8 elsewhere Detere a b ad c. Suluto a The s obtaed b tegtatg over all at each value o. or ad because beg ero leads to a ero tegral. For s detered as I suar ca be wrtte as d 8d b s obtaed b tegtatg over all at each value o. For ad the because s ero leads to a ero tegral. For s detered as Thus ca be wrtte as d 8d c The codtoal dest ro.6 or ad s 8 For the ot dest s ero ad thus ca be suared as elsewhere da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

34 Rado Sgal Processg Chapter Rado Varables Several plots o or varous values o are show Fgure.9. Notce the deret doas or oero values. aple.7 or the rado varables ad that have the ot probablt dest ucto gve Fd aple.6. Soluto Usg the codtoal dest detered aple.6 ad.9 the codtoal epected value s detered or as d d For outsde the terval the dest s ero as s the epected value. Two rado varables ad are deed to be statstcall depedet the ot probablt dest ucto ca be wrtte as a product o the two argal destes or all ad as show:.96 The correlato betwee two rado varables ad s deed as the epected value o ther product; R.97 Two rado varables are deed as ucorrelated R whch s the sae as wrtg.98 Usg q..96 t s sple to show that two rado varables are statstcall depedet the the are ucorrelated. The steps the proo ollow. dd dd d d.99 Two rado varables ad are called orthogoal. I the rado varables ad are depedet ad both o the eas are ero or oe o the eas s ero ad the other s te the the are orthogoal sce b depedece the ad thus. These relatoshps betwee depedece orthogoalt ad ucorrelated rado varables are show Fgure.. I the gure the sgle sold arrow dcates alwas true whereas a dotted arrow dcates alwas true the codtos wrtte alog sde are true. The arrow bac ro ucorrelated to depedece wll be show later to be true or otl Gaussa rado varables. da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

35 Rado Sgal Processg Chapter Rado Varables Jotl Gaussa Idepedet Alwas Ucrrelated Oe ea ero & other ea te Orthogoal Oe ea ero & other ea te Fgure. Relatoshp betwee the detos o depedet ucorrelated ad orthogoal. Wth two rado varables the dvdual varaces are as beore or each rado varable but ore orato cosderg the relatoshp betwee ad s avalable through the covarace deed as dd. The correlato coecet oraled covarace b betwee the two rado varables ad s deed as the. It provdes a easure o lear relatoshp betwee the two rado varables ad. the correlato coecet or two rado varables ad ca be easl show to be bouded b ad as The closer. s to or the ore the rado varables ad are sad to be learl related. It s otced that the closer s to or the ore rdgele the probablt dest becoes whch dcates that oe rado varable ca be wrtte as alost a scaler ultple o the other. The hgher-order oets are deed b ad the cetral oets or two rado varables ad dd. dd. da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 5 o 58

36 Rado Sgal Processg Chapter Rado Varables da Uverst Lu Cogeg Page 6 o Jotl Noral Rado Varables The rado varables ad are deed as otl Gaussa or otl oral ther probablt dest ucto has the ollowg or: ep r r.5 Where ad are respectvel the ea o ea o varace o varace o ad correlato coecet o ad ad r. The argal destes or destes o the rado varables ad ca be obtaed b tegratg out the proper varable to gve } ep{ } ep{.6 B dvdg the ot dest b the argal dest ad splg the codtoal probablt dest ucto or codtoal o ad codtoal o the codtoal probablt dest ucto ca be detered as } ep{ } ep{.7 Whe the eas are ero the codtoal eas ad varace are easl obtaed ro the above b settg to gve } ep{ } ep{.8 Note that although ad have ero eas the codtoal eas are ot equal to eros. A uber o useul orulas or varous oets o otl oral ero ea rado varable ters o ther gve paraeters ad ollow:

37 Rado Sgal Processg Chapter Rado Varables Where s odd cos s.9.7 Two Fucto o Two Rado Varables The basc proble s to obta the statstcal characterato o two rado varables Z ad W that are uctos o two other rado varables ad. The epressos or Z ad W are speced b g ad Z h W..7. Probablt Dest Fucto Dscrete Rado Varables I ad are dscrete rado varables ther ot probablt dest ucto s coposed o two-desoal pulses: p. The the ot probablt dest ucto or W g ad Z h pulses wth locatos obtaed b tag wll also cota ol through the ucto w g ad h ad wth weghts equal to the probabltes o each. The w s thereore see to be WZ WZ w p w g h. aple.8 Gve the ollowg probablt dest ucto or the dscrete rado varables ad : 6 Cosder two ew rado varablesw ad Z that are deed b the ollowg trasoratos: W Z Fd the ot probablt dest ucto WZ w. da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 7 o 58

38 Rado Sgal Processg Chapter Rado Varables Soluto Tag each delta ucto through the trasorato gves the ot dest w as WZ w w w w 6 w w w WZ w.7. Probablt Dest Fucto Cotuous Rado Varables ad Cotuous Fuctos I ad are cotuous rado varables ad g ad h are cotuous uctos o ad wth o lat spots the couterpart to the oe ucto o oe rado varable result ca be wrtte as ollows or rado varables W g ad Z h Where WZ w. J are all real solutos o w g ad h. The J s the Jacoba o the trasorato whch s epressed as the ollowg deterat: g g J. h h I o real soluto o w g ad h probablt dest WZ w or those values o w ad. ests or a gve w ad the the ot aple.9 Let W ad Z be two rado varables deed b the ollowg uctos o two other rado varables ad : W adz Let ad be depedet Gaussa rado varable descrbed b ~ N ad ~ N. a Fd the ot probablt dest ucto or the rado varables ad. b Are ad statstcall depedet? Soluto a The soluto s obtaed b usg qs.. ad.. The Jacoba or the trasorato deed s da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 8 o 58

39 Rado Sgal Processg Chapter Rado Varables J We ust d all real solutos o the equatos w ad or all w ad. For w ad the secod equato ca be solved or ters o as. Substtutg ths result to the rst equato allows us to solve or as w be real. Sce wll ot w there s o real soluto or w so b the trasorato theore the ot dest WZ w. For there are o real solutos or a w or sce w ad ca ever be egatve thus WZ w or those regos o w ad. I suar there are ol real roots or w ad w ad or that case there are our o the as cobatos o the ollowg: w. B q.. the ot dest or w ad becoes WZ w w w w w Substtutg these roots to the ollowg dest ucto ep ad collectg le ters the are all the sae because o the squarg operato the uerator ad the Gaussa dest the w s easl see to be WZ WZ w e w w w Z w b The rado varablesw ad Z are ot statstcall depedet because the ot dest s ot a product o the argal destes. Ths ca be easl see b dg the argal destes ad ultplg the. However t s easer to see that the product caot be the sae as the product would be oero or all w ad whereas the ot dest s ero or w ad w. aple. Let Z ad W be rado varables deed b the ollowg uctos o two other rado varables ad : da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 9 o 58

40 Rado Sgal Processg Chapter Rado Varables W Z Assue that s ow ad that t s requred to d the ot dest ucto WZ w. Soluto The trasorato theore caot be used sce the Jacoba s gve b J regardless o the roots o w ad. CertalW ad Z are rado varables so the ust have a probablt dest ucto. Notce that Z W that s Z ca be wrtte as a lear ucto ow. Thus asw taes o values w Z taes o ol the values w. I the w two-desoal space the ot dest ust be ero or w ot o the le ad have eag alog the le w as show Fgure.. Ths s what s eat b a le ass. Usg the dstrbuto ucto ethod the dervatve elds the ot dest ucto as w w w WZ W where W w s obtaed b usg the oe ucto o two rado varables approach..7. Dstrbuto Fucto Cotuous Dscrete or Med The proble o dg the ot cuulatve dstrbuto ucto or the rado varables W ad Z deed b F WZ w ca be wrtte as F WZ g Z h W.7 w P{ W w Z } P{ g w h }.8 The P{ g w h } ca be obtaed b tegratg the probablt dest ucto over the rego I deed the - space as WZ I WZ { : g w ad h }.9 Usg ths I ad F WZ w ca ow be wrtte as WZ F WZ w dd. I WZ Ths approach ca be used regardless whether the probablt dest ucto s ro a dscrete cotuous or ed rado varable. I WZ w s desred t ca be obtaed b tag partal dervatves as ollows: da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

41 Rado Sgal Processg Chapter Rado Varables WZ w FWZ w. w Specal care s requred F WZ w s dscotuous sce varous tpes o pulse uctos wll be geerated. aple. Dee two rado varablesw ad Z b the ollowg trasoratos o two rado varables ad : W ad Z. Assue that ad are depedet rado varables charactered b ther ot probablt dest ucto whch s gve b e. Fd the ot cuulatve dstrbuto ucto w orw ad Z ad the tae partal dervtves to obta F WZ the ot probablt dest ucto WZ w. Soluto Usg q.. to obta the cuulatve dstrbuto ucto depeds o rst obtag the rego I WZ whch or our proble s deed b ad s show Fgure.6. I WZ : wad w w= + I WZ + W W For w ad I the ull set. Clearl the tegral over that rego gves ero ad so F WZ w. Fgure.6 The rego o tegrato I WZ or aple. WZ For w ad the rego s show as the shaded area Fgure.6. Thus the w ca be wrtte ters o the tegral o the ot probablt dest ucto or the rado varables ad F WZ da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

42 Rado Sgal Processg Chapter Rado Varables da Uverst Lu Cogeg Page o 58 as ollows: w w w WZ d d e e d d w F ep Ths epresso ca be spled to gve w d e e w F w WZ The ot probablt dest ucto w WZ s the obtaed b tag the partal dervatves o w F WZ wth respect to w ad as ollows: w e e w w d e e w w F w w w w WZ WZ I the equato above the partal wth respect to was obtaed b Lebt s rule. Cotug ad tag the partal wth respect to w ad splg gves w w e e w w WZ Ths result ca be easl checed b usg the trasorato theore or two uctos o two rado varables..8 Oe Fucto o Two Rado Varables Dee a rado varable Z b g Z where ad are rado varables. The basc proble s that o characterg Z owg ether partal or ttal characteratos o ad..8. Probablt Dest Fucto Dscrete Rado Varables I ad are dscrete rado varables ther ot probablt dest ucto s coposed o two-desoal delta uctos as P. Where } { P P ad S s the set o all pars cosdered. I g

43 Rado Sgal Processg Chapter Rado Varables represets a ucto deed o real ad the the rado varable Z g s a dscrete rado varable wth probablt dest ucto Z gve b Z S P g Soe o the g wll be the sae so addg those probabltes the Z ca be wrtte ters o a reduced uber o ters. I s a eleet ro the set o uque values o g where S the Z ca be rewrtte as Z P. K Where P P{ : g } ad K s the set o all uque value o g..8. Probablt Dest Fucto Cotuous Rado Varables Let ad be cotuous rado varables wth ow ot probablt dest ucto ad g be a ow cotuous real-valued ucto o ad. The basc proble s that o deterg the probablt dest ucto Z or the rado varable deed b Z g. Four portat basc ethods or dg the dest are the cuulatve dstrbuto approach the aular rado varable approach the creetal probablt approach ad the sthetc saplg or Mote Carlo approach. O these ol the sthetc saplg approach would be cosdered a drect approach. Cuulatve Dstrbuto approach. I ths approach the cuulatve dstrbuto ucto F Z s oud ad the deretated to obta Z. The dstrbuto ucto F Z s detered as ollow: g F P Z P dd.5 Z Where { : g }. To obta F Z or ro to ust be I tegrated over chagg regos o tegrato. The regos are coposed o all such that g wth boudares equal to the tersecto o a plae at ad the ucto as show Fgure.7. The probablt dest ucto Z s the obtaed b deretatg F Z to gve d F.6 d For certa probles whose regos ca be epressed easl the lts the two-desoal tegral Lebt s rule ca be appled so that the tegral a ever eed to be evaluated. However or other probles the ethod s a tour de orce usg two-desoal tegrato. I da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

44 Rado Sgal Processg Chapter Rado Varables g= I Fgure.7 Obtag the rego o tegrato or the dstrbuto ethod. aple. Let ad be a par o rado varables charactered b ther ot probablt dest ucto. Dee a ew rado varable Z b Z. Fd the probablt dest ucto Z or Z b usg the dstrbuto ethod ad detere the aswer or the specal case where ad are depedet rado varables. Soluto To d the dstrbuto ucto F Z the regos I ust be deted or all ad are gve b I Z : Thus the rego s see to be all pots below ad cludg the boudar le or stadard or as see Fgure.8. Thus we ca wrte F Z or a arbtrar as the ollowg tegral: FZ dd The probablt dest ucto s the gve b d dd d Iterchagg the rst tegral ad the dervatve ad applg Lebt s rule to the secod tegral gves the aswer as d.7 da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page o 58

45 Rado Sgal Processg Chapter Rado Varables =- I Z {+} Fgure.8 Rego I Z or aple. I ad are depedet rado varables the. For ths portat case the Z or the rado varable Z covoluto tegral as ollows: s detered ro q..7 to be the d.8 Aular Rado Varable Approach. The aular rado varable approach s a ethod or obtag the probablt dest ucto Z h. The approach cossts Z or the rado varable three basc steps: deg a aular rado varable usg the thasoratoal theore to obta a ot probablt dest ucto ad a tegrato to obta the argal dest or Z ad thus the desred dest. I the rst step a rado varable W s deed b W g ths s a cttous rado varable ad selected as a vehcle or usg the two-desoal trasorato theore. Usuall g s selected as ust ust or a sple ucto o ad that ght be suggested b the or o h whch dees Z. the ucto The secod step s the applcato o the two-desoal trasorato theore to the ollowg: W g.9 Z h To d the ot probablt dest WZ w whch s gve b Where WZ w. J are all real soluto o w g ad h. The al step the process o obtag the probablt dest ucto Z s to tegrate the da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 5 o 58

46 Rado Sgal Processg Chapter Rado Varables ot dest ucto WZ w as ollows: Z WZ w dw. aple.5 Let ad be a par o rado varables charactered b ther ot probablt dest ucto. Dee a ew rado varable Z b Z sae as aple..fd the probablt dest ucto Z or b usg the aualar rado varable ethod. Soluto Dee the aular rado W whch ow gves us two uctos o two rado varables W ad Z. Usg the two uctos o two rado varables theore the ot probablt dest WZ w s gve b q... There s ol oe soluto to the set o equatos w ad whch are easl see to be w ad w. The Jacoba o the trasoratos s detered as g g J h h Thereore the ot dest ca be wrtte as WZ w w w w w The desred probablt dest ucto W w WZ w to gve the result w dw w w dw WZ s obtaed b tegratg the ot dest Icreetal Approach. The creetal approach ca be used or certa probles o dg the probablt dest ucto Z h. The ethod uses the act that Z Where P{ Z } I Z or rado varable Z deed b ca be approated b Z P{ Z }. s equvalet to the probablt that { : h }. Thus the approato above ca be wrtte as s a eber o dd. Z I da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 6 o 58

47 Rado Sgal Processg Chapter Rado Varables The approach reles o the assupto that as the tegral over I becoes a lear ucto o. Whe ths happes the Z the lt as. s caceled ro both sdes o the approato thus leavg aples.6 Suppose we are gve rado varables ad that are charactered b ther ot probablt dest ucto ad that or. A ew rado varable Z s deed b Z the product o the two rado varables. Fd the probablt dest ucto Z or Z b the creetal ethod. Soluto Usg the creetal ethod we ust rst detere the probablt that Z s the terval whch s gve b P Usg. gves Cacellg the Z P / / Z P d dd d the equato above the lt reduces to the result Z Ths proo ca be oded to clude the case where or to gve the ollowg result or the probablt dest Z or the product o a two rado varables Z : d d. Z.9 Coputato o h As or the case o oe rado varable there are several basc ethods or obtag Z h Z where. The rst three ethods descrbed are coparable to those descrbed or oe ucto o oe rado varable. The orth techque s ver useul or uctos o ore tha oe rado varable ad does ot have a couterpart the ucto o oe rado varable case. Method. Usg da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 7 o 58

48 Rado Sgal Processg Chapter Rado Varables Method. Usg Z Z h h dd.5 h Z Z d.6 Method. Usg a Mote Carlo techque or sthetc saplg. Method. Usg terated epected value Z h h h.7 Method has o parallel or the case o oe ucto o oe rado varable but ca be a powerul ethod or coputg epected values o uctos o two rado varables. The eag o terated epected value operato descrbed above s easl uderstood ad represets codtoal epected values as ollows: h h dd.8 Breag the ot probablt dest ucto to the product o ts codtoal ad argal destes gves. h h dd.9 Rearragg the tegral o the rght sde o the equato above ad tag the through the tegral wth respect to t s ot a ucto o gves h h d d. The ter bracets s recoged as the codtoal epected value o Thereore h becoes I a slar asho h as ollows g or h h d. h h d. ca be wrtte ters o the codtoal epected value wth respect to Z h h. da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 8 o 58

49 Rado Sgal Processg Chapter Rado Varables aple.7 Gve Z cos wth a uor rado varable o ad a Gaussa rado varable wth ero ea ad varace equal to. Assue that ad are depedet ad copute Z the epected value o Z usg ethod. Soluto Usg.7 we have the epected value o Z as where Z cos cos cos cos d For ths proble ad are depedet thereore the codtoal dest s the sae as the argal dest ad the equato above ca be wrtte as The epected value o Z ca be oud cos d cos Z d The ollowg eaple calculates soe hgher-order oets or otl Gaussa rado varable also usg the cocept o terated epected values. aple.8 Let ad be otl oral rado varables wth paraeters wth ad ow. Fd ad. Soluto Usg the terated epected value orula ust preseted the ca be wrtte as where dd ep The ter sde the square bracets ca be spled b tag through the tegral ad usg the result or the codtoal ea q..8 to gve d d Substtutg ths result or the braceted ter ad usg secod oet or the rado varable gves the as da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 9 o 58

50 Rado Sgal Processg Chapter Rado Varables Slarl ca be oud the ollowg asho: d d d d d d The tegral the paretheses above represets the codtoal secod oet ad t ca be wrtte ters o the codtoal varace ad the codtoal ea or q..7 as d d The rst tegral s the varace o sce has a ero ea ad the secod s the ourth-order oet or the oe-desoal Gaussa rado varable whch s ow ro q... Substtutg these secod- ad ourth-order oets gves d. Multple Rado Varables.. Total Characteratos The real-valued uctos or rado varables ad are deed o a saple space ad are sad to be totall charactered or descrbed wth relato to calculatg probabltes o acceptable evets.e. evets a eber o F b ther cuulatve dstrbuto ucto or probablt dest ucto. The ot cuulatve dstrbuto ucto F or rado varables s deed b F P{ }. It satses the etesos o the bouded odecreasg ad cotut propertes gve.6. ad ca da Uverst Lu Cogeg -Mal:clu@al.da.edu.c Page 5 o 58

51 Rado Sgal Processg Chapter Rado Varables da Uverst Lu Cogeg Page 5 o 58 be used to detere probabltes o gve evets. The ot probablt dest ucto s related to the ot cuulatve dstrbuto ucto through partal dervatves ad vce versa ters o tegrals as d d d F F.5 For a gve vector o rado varables the characterstc ucto s deed as the oded Fourer trasor o the ot probablt dest ucto ω ω F d d d e e.6 Where T ω. Ths tegral does ot ecessarl est or all probablt dest uctos ad has a gve rego o covergece. The oet-geeratg ucto s deed as ollows: t t t t t t d d d e e t t t M.7.. Partal Characteratos Partal characteratos clude the varous argal ad codtoal destes ad the hgher-order oets. The hgher-order oets ca be detered ro the deto o the oets. However a probles the characterstc ucto or oet-geeratg ucto s useul or that purpose. The rth-order hgher-order oets or r are deed b d d d.8 Where r A well-ow theore gves the hgher rth-order oets ters o the characterstc ucto as ollows:

52 Rado Sgal Processg Chapter Rado Varables da Uverst Lu Cogeg Page 5 o 58 r r.9 Where r ad r Slarl the oets ca be geerated b usg the oet-geeratg ucto through a correspodg theore ro Daveport that gves the hgher-order oets ters o the oet-geeratg ucto as ollows: t t t r t t t t t t M.5 Where r ad r The oet-geeratg ucto s spler to obta that t s detered as a real tegra. However the use o the estg Fourer trasor table to obta the characterstc ucto ad the act the characterstc ucto ests or cases where the oet-geeratg ucto does ot est aes the characterstc ucto ore usable. aple.9 The rado varables ad are charactered b ther ot probablt dest ucto whch s 6 e Detere b usg q..9 the ollowg thrd-order oets: a b c d. Soluto To obta the oets ro.9 we rst eed to calculate the characterstc ucto whch ro.6 s as ollows: 6 6 d d d e e a The rst-order oet s detered ro.9 as 6 b The thrd-order oet s detered ro.9 as 6 6