Chapter 2: Random Variables

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1 Chapte : adom aiables.. Cocept o a adom aiable.. Distibutio Fuctios.3. Desity Fuctios Fuctios o adom aiables.4. Mea alues ad Momets Hypegeometic Distibutio.5. The Gaussia adom aiable Histogams.6. Desity Fuctios elated to Gaussia.7. Othe Pobability Desity Fuctios.8. Coditioal Pobability Distibutio ad Desity Fuctios.9. amples ad Applicatios Aoucemet: SW Hosted Idusty Die ad etwokig, Febuay! Fo those o you who did ot get the iitatio o ay o the gieeig Facebook pages, the Society o Wome giees is hostig a idusty die o all egieeig studets Thusday, Febuay th, 5 om 6:3-8:. Seeal compaies (ato, Whilpool, Comsumes egy, etc.) ae comig to etwok with OU oe a ice, ee die. Thee will also be a ew key speakes to talk about thei epeieces i the wokoce. Seats will be gie o a ist come ist see basis. I you would like to egiste o this eet please you esume i pd om to swe.idustydie.wmu@gmail.com. I look owad to seeig you at the die! Chelsea ussell SW Idusty Die Coodiato Weste Michiga Uiestiy otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 o C 38

2 -5 Gaussia o omal pobability desity uctio The Gaussia o omal pobability desity uctio is deied as:, ep o whee is the mea ad is the aiace. The omal desity uctio has a zeo mea ad uit aiace. See Appedi D, p ot yet poe easos o impotace:. It poides a good mathematical model o a geat may dieet physically obseed adom pheomea that ca be justiied theoetically i may ways.. It is oe o the ew desity uctios that ca be eteded to hadle a abitaily lage umbe o adom aiables coeietly. 3. Liea combiatios o Gaussia adom aiables lead to ew adom aiables that ae also Gaussia. This is ot tue o most othe desity uctios. 4. The adom pocess om which Gaussia adom aiables ae deied ca be completely speciied, i a statistical sese, om a kowledge o the ist ad secod momets. This is ot tue o othe pocesses. All highe leel momets ae sums, poducts ad/o powes o the mea ad aiace. 5. I system aalysis, the Gaussia pocess is ote the oly oe o which a complete statistical aalysis ca be caied though i eithe the liea o oliea situatio. 6. The uctio is iiitely dieetiable (all the deiaties eist). otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 o C 38

3 Gaussia o omal Distibutio To summaize, hee is a list o situatios whee appoimate omality is sometimes assumed. Fo a ulle discussio, see below. I coutig poblems (so the cetal limit theoem icludes a discete-to-cotiuum appoimatio) whee epoductie adom aiables ae ioled, such as Biomial adom aiables, associated to yes/o questios; Poisso adom aiables, associated to ae eets; I physiological measuemets o biological specimes: The logaithm o measues o size o liig tissue (legth, height, ski aea, weight); The legth o iet appedages (hai, claws, ails, teeth) o biological specimes, i the diectio o gowth; pesumably the thickess o tee bak also alls ude this categoy; Othe physiological measues may be omally distibuted, but thee is o easo to epect that a pioi; Measuemet eos ae assumed to be omally distibuted, ad ay deiatio om omality must be eplaied; Fiacial aiables The logaithm o iteest ates, echage ates, ad ilatio; these aiables behae like compoud iteest, ot like simple iteest, ad so ae multiplicatie; Stock-maket idices ae supposed to be multiplicatie too, but some eseaches claim that they ae Ley-distibuted aiables istead o logomal; Othe iacial aiables may be omally distibuted, but thee is o easo to epect that a pioi; Light itesity The itesity o lase light is omally distibuted; Themal light has a Bose-istei distibutio o ey shot time scales, ad a omal distibutio o loge timescales due to the cetal limit theoem. otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 3 o C 38

4 The Gaussia Pobability Distibutio Fuctio is F ep d The PDF ca ot be epeseted i a closed om solutio! Impotat otes o the cue:, ep o. Thee is oly oe maimum ad it occus at the mea alue.. The desity uctio is symmetic about the mea alue. 3. The width o the desity uctio is diectly popotioal to the stadad deiatio,. The width o occus at the poits whee the height is.67 o the maimum alue. These ae also the poits o the maimum slope. Also ote that: P 683 P The maimum alue o the desity uctio is iesely popotioal to the stadad deiatio,. 5. Sice the desity uctio has a aea o uity, it ca be used as a epesetatio o the impulse o delta uctio by lettig appoach zeo. That is lim ep otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 4 o C 38

5 otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 5 o C 38 Momets ad Cetal Momets o Gaussia ee odd, 3, 4, 6 3 3, 3,, This is a patial list. As peiously stated all momets ae a uctio o the mea ad aiace/stadad deiatio. As a coeiet popety, the Gaussia is completely descibe oce the mea ad aiace ae kow!

6 Computig the Gaussia PDF The PDF is tabulated i Appedi D o a zeo mea, uit aiace pd. Fo these alues, it is ote descibed as omalized ad is deied as d ep The distibutio uctio is the coeted based o the elatioship u F u Whe usig Appedi D, the egatie alues o u ae deied om the positie as u u The omal uctio is peectly symmetic about zeo; theeoe, this popety must eist. The tail o the uctio om iiity to u must be equialet to the tail o the uctio om +iiity to u..9 Gaussia omal Desity ad Distibutio : :.843 : :.977 pd PDF : :.5 : : : - :.4 : : otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 6 o C 38

7 Aothe deied uctio that is elated to the Gaussia (ad used) is the Q-uctio, this is the pobability om the tail o the Gaussia see Appedi.: Q u ep du u The Q-uctio is the complemet o the omal uctio, : Q Theeoe ote that: Q Q F Q Gaussia omal Desity, Dist, ad Q Fuctio pd omal Fuctio Q Fuctio otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 7 o C 38

8 A appoimatio o the Q uctio (poided i the tet is) p. 7 ep, o a 3 Q a a a a The tails o the Gaussia ad theeby the Q-uctio ae used o eets that happe ey aely. Tet eample: False detectio i a IC tigge cicuit o detectig digital sigal leels. Assume Gaussia oise o zeo mea ad.^ aiace. Zeo leel:.5 plus oise Oe leel detectio:.5. Ca o will ad at what pobability will the zeo ceate a alse oe? Zeo-leel equialet: Gaussia with.5 mea ad.^ aiace. P Q Q App. So, i a ew msec o detectio leel o a gigahetz clocked system say, clocks what is the pobability thee has bee a alse tigge? P FalseTigg e P.5. 3 otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 8 o C 38

9 Aothe way to id alues o the Gaussia The eo uctio, deied as F e ep u du Q u e e e The eo uctio ( = F()) is built-i to MATLAB. Appedi G poides a oeiew o the uctios ad how they ca be used withi MATLAB. ote: Thee is a typo i the deiitio o the e o p. 44. The itegatio should be as show aboe, ot the itegal om to iiity. Fom MATLAB: F o uctio. = F() is the eo uctio o each elemet o. must be eal. The eo uctio is deied as: e() = /sqt(pi) * itegal om to o ep(-t^) dt. See also ec, ec, ei. eeece page i Help bowse doc e Cetal Limit Theoem The omalized sum o a lage umbe o idepedet aiables, haig the same pobability desity uctio, has a pobability desity uctio that appoaches a Gaussia desity uctio. k k k ad k otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 9 o C 38

10 ample: Itelligece Quotiet (IQ) Fo IQ Tests: IQ ad IQ 5 Fo a IQ test you e take, the paticula test has a aw scoe mea ad stadad deiatio o 5 ad 4 I you scoe was 8, what uppe pecet ae you a membe o (i.e., you would be i the top %)? aw aw The taslatio acto o the omal distibutio ioles: z Fist id the omal distibutio alue descibig you peomace: Fom the table i appedi D F To detemie the uppe pecetile 8 5 z F What would be you equialet scoe o a poit test? 8 5 z IQ IQ IQ IQ z IQ IQ otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 o C 38

11 Histogams Poidig a meas o showig the esults o epeated epeimets match the epected pobability desity uctio (o pobability mass uctio) that was used (o epected). The pd F u du The pm F u otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 o C 38 pd pm u To taslate om a cotiuous distibutio to a bi-based discete distibutio, take the aea o the desity cue withi the age o a bi that you wish to deie ad equate it to the pobability mass uctio at the cete o the bi Let be diided ito bis so that bi i ages om i bi bi bi i i bi Fom pd B k B i F i p u du k B k B B k B pm k u Let du B k B i pm k The F i k What would a ist ode appoimatio o the pm elemets be? The bi cete times the bi width. B k B pm B k B u du k B B k

12 Tuig a distibutio ito bis: Assume that a adom aiables pobability desity uctio is tiagula ad ca be descibed as pd,,,, o o o o Use 5 bis located. The bi width is /5=.4 The bi cetes ae at.8, -.4,.,.4, ad.8. Compute the pm elemets..6.6 u.36 pm u du u pm. u u du u pm pm 3. u du. u u du u pm 4. 4 pm. u u. pm 5. 8 pm A a-pioi pm deied om the pd. ow o a epeimet i I a epeimets, how may esults would I epect to eside i each bi? * pm i otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 o C 38

13 Compute amples See GaussiaGe.m Gaussiaad TiGe.m Tiad.m poetialge.m I you kow the PDF, you ca use a iese pocess to geeate adom alues. F The F otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 3 o C 38

14 ample 4. Biay Commuicatios Fom: Albeto Leo-Gacia, Pobability, Statistics, ad adom Pocesses Fo lectical gieeig, 3d ed., Peaso Petice Hall, 8, ISB: A biay tasmissio system seds a bit by tasmittig a oltage sigal, ad a bit by tasmittig a +. The eceied sigal is coupted by Gaussia oise ad gie by: whee is the tasmitted sigal, ad is a oise oltage with pd P[ ] = p = P[ ]. Fid the pd o.. Assume that Let B be the eet is tasmitted ad B be the eet is tasmitted, the om a patitio, ad F F B P B F B P B Sice the eet F P p P p the eet is equialet to ad is equialet to F F B P F B P F. Theeoe the coditioal cd s ae, ad The cd is the The pd is the F F p F p d d F d d d d F p F p p p The Gaussia adom aiable has pd: ep, o otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 4 o C 38

15 otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 5 o C 38 The coditioal pds ae ep B ep B The pd o the eceied sigal is the: p p ep ep A detectio theshold, TH, ca be established based o the pd. The, the pobabilities o coectly detectig the tasmitted sigals ca be computed. as >TH ad <TH..6 Desity Fuctios elated to Gaussia ayleigh Distibutio D Gaussia Mawell Distibutio 3D Gaussia

16 ayleigh Distibutio Fo a two dimesioal poblem (positios i ad y with two idepedet Gaussia adom aiable oise o oset tems), the distace om a desied poit is descibed as a adial o ecto magitude, the adial eo o oset is descibed by the ayleigh Distibutio. Fo The pobability desity uctio (pd) is, ep, o o The pobability distibutio uctio (PDF) ca be deied as: ep F d,, F ep o o, o, o F ep, o, o otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 6 o C 38

17 otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 7 o C 38 The ist momet ep d Closed om solutio (p. 4) 3 4 ep a d a 4 3 The secod momet ep d Math tables hit u u u u u ep d u ad ep u ad ep ad keep goig util you get. The secod cetal momet, aiace o stadad deiatio is

18 ample: Achey taget shootig with.5 4 Assume a oot adius taget with a ich adius Bulls-eye ep, o, o ep F,, o o F 6 ep 8,, ep 8,, o o o o 3 ep d Pobability o a Bulls-eye ( ich adius) F ep ep Pobability o missig the taget ( oot adius) F 4 ep 8 ep otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 8 o C 38

19 otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 9 o C 38 Mawell Distibutio Fo a thee dimesioal poblem (the elocity o a paticle i a peect gas), the elocity ecto magitude is descibed by the Mawell Distibutio. Fo Z The pobability desity uctio (pd) is,, ep 3 o o The pobability distibutio uctio (PDF) ca be deied as:,, ep 3 o o d F The solutio equies umeical itegatio! The ist momet 8 ep 3 d The secod momet 3 ep d 3 The secod cetal momet, aiace o stadad deiatio is

20 otes ad igues ae based o o take om mateials i the couse tetbook: Pobabilistic Methods o Sigal ad System Aalysis (3d ed.) by Geoge. Coope ad Clae D. McGillem; Ood Pess, 999. ISB: B.J. Bazui, Spig 5 o C 38

MATH Midterm Solutions

MATH Midterm Solutions MATH 2113 - Midtem Solutios Febuay 18 1. A bag of mables cotais 4 which ae ed, 4 which ae blue ad 4 which ae gee. a How may mables must be chose fom the bag to guaatee that thee ae the same colou? We ca