Chapter 2: Random Variables

Transcription

1 Chp : ndom ibls.. Concp of ndom ibl.. Disibuion Funcions.. Dnsiy Funcions Funcions of ndom ibls.. n lus nd omns Hypgomic Disibuion.5. h Gussin ndom ibl Hisogms.. Dnsiy Funcions ld o Gussin.7. Oh obbiliy Dnsiy Funcions.8. Condiionl obbiliy Disibuion nd Dnsiy Funcions.9. mpls nd Applicions Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 of 8 C 8

2 Homwo oblm qus/discussion -7. (unifom dnsiy) A coninuous-lud ndom olg nging fom - o + is o b qunizd so h i cn b psnd by biny squnc..) If h ms qunizing o is o b lss hn % of h mimum lu of h olg, find h minimum numb of qunizing lls h quid. q h inc fo unifom disibuion is. q h ms olg siz is hn q ms qunizion o % of m olg implis. q..98 h numb of quid lls is hn q ounding up would qui 58 lls. Ys his is sng wy o s h poblm nd find soluions! b.) If h numb of qunizing lls is o b pow of, find h minimum numb of qunizing lls h will sill m h quimns. h n highs pow of is o ^. c.) How mny biny digis quid o psn ch qunizing ll? o psn h lus, you nd -bi biny numb. olg sp sizs.5 q [Finding n ADC] Qunizion o is +/- ½ ls-significn bi. Fo % ccucy (o b), I would consid (in biny) 8 lls ( 7 ) o mo. o dquly co +/- ng, I would m h ADC msu +/- 5. ng nd hn h biny olg sps of.. h mn o would b zo, h mimum o would b +/-. nd h ms o would b sq(. /) =.58. Sndd ADC numb of bis: 8,,,, nd ypicl. Impon fco in slcion, wh smpl is quid! h yp nd cos will y in lionship o smpls-p-scond. Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 of 8 C 8

3 . Dnsiy Funcions ld o Gussin yligh Disibuion Fo wo dimnsionl poblm (posiions in nd y wih wo indpndn Gussin ndom ibl nois o offs ms), h disnc fom dsid poin is dscibd s dil o co mgniud, h dil o o offs is dscibd by h yligh Disibuion. Fo X Y h pobbiliy dnsiy funcion (pdf) is f, p, fo fo h pobbiliy disibuion funcion (DF) cn b did s: p F d,, fo fo F p, fo, fo yligh Dis. nd Dnsiy Fundions pdf DF Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 of 8 C 8

4 Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 of 8 C 8 h fis momn p d Closd fom soluion (p. ) p d h scond momn p d h bls hin u u u u u p d u nd p u nd p nd p going unil you g. h scond cnl momn, inc o sndd diion is

5 mpl: Achy g shooing wih Gussin X Y.5 Assum foo dius g wih inch dius Bulls-y f p, fo, fo p F,, fo fo f F p 8,, p 8,, fo fo fo fo p d.. 8 obbiliy of Bulls-y ( inch dius) F. p 8 obbiliy of missing h g ( foo dius) s ylighgn.m F s yligh.m (bsd on boo mpl p. 9) 8 p p 8 p 8.5 Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 5 of 8 C 8

6 Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 of 8 C 8 wll Disibuion Fo h dimnsionl poblm (h lociy of picl in pfc gs), h lociy co mgniud is dscibd by h wll Disibuion. Fo Z Y X h pobbiliy dnsiy funcion (pdf) is,, p fo fo f h pobbiliy disibuion funcion (DF) cn b did s:,, p fo fo d F h soluion quis numicl ingion! h fis momn 8 p d h scond momn p d h scond cnl momn, inc o sndd diion is 8.5 8

7 s wllgn.m fom hp://n.wiipdi.og/wii/wll-bolzmnn_disibuion wll-bolzmnn Dis. nd Dnsiy Fundions pdf DF Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 7 of 8 C 8

8 Oh Disibuions ponnil Disibuion Fo ns h li o n ponnil dcy, such s h pobbiliy of nw n hppning in h n m. h pobbiliy dnsiy funcion (pdf) is f, p, fo fo h pobbiliy disibuion funcion (DF) cn b did s: F p, fo, fo hp://n.wiipdi.og/wii/ponnil_disibuion Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 8 of 8 C 8

9 Fom hp://n.wiipdi.og/wii/ponnil_disibuion In l wold scnios, h ssumpion of consn (o pobbiliy p uni im) is ly sisfid. Fo mpl, h of incoming phon clls diffs ccoding o h im of dy. Bu if w focus on im inl duing which h is oughly consn, such s fom o duing wo dys, h ponnil disibuion cn b usd s good ppoim modl fo h im unil h n phon cll is. Simil cs pply o h following mpls which yild ppoimly ponnilly disibud ibls: h im unil you h you n c ccidn; h im unil dioci picl dcys, o h im bwn bps of gig coun; h numb of dic olls ndd unil you oll si ims in ow; h im unil lg mo si cuss mss incion n. ponnil ibls cn lso b usd o modl siuions wh cin ns occu wih consn pobbiliy p uni disnc: h disnc bwn muions on DNA snd; h disnc bwn od ill on gin highwy; In quuing hoy, h in-il ims (i.. h ims bwn cusoms ning h sysm) ofn modld s ponnilly disibud ibls. libiliy hoy nd libiliy ngining lso m nsi us of h ponnil disibuion. Bcus of h mmoylss popy of his disibuion, i is wll-suid o modl h consn hzd poion of h bhub cu usd in libiliy hoy. In physics, if you obs gs fid mpu nd pssu in unifom giionl fild, h highs of h ious molculs lso follow n ppoim ponnil disibuion. his is consqunc of h nopy popy. Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 9 of 8 C 8

10 h fis momn p d Fom bls p p d p h scond momn p d h scond cnl momn, inc o sndd diion is h mmoylss popy: Di h h h Bsd on pio s hoy h h h h h h h W now h h F h h h p p hfo Bu his is h sm s p h h h p p h, h pobbiliy is no dpndn upon h im hisoy nd hby cn b clld mmoylss. (No, his is h only coninuous ndom ibl h hibis his popy.) Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 of 8 C 8

11 mpl: Componn filu Componn filu in spccf occus indpndnly nd unifomly wih h g im bwn filus of dys. h spccf ss on -dy mission wih ll componns funcioning. Wh is h pobbiliy h i will compl h mission wihou componn filu? nd h pobbiliy dnsiy funcion (pdf) is f p,, fo fo h pobbiliy disibuion funcion (DF) cn b did s: F p, fo, fo obbiliy of no filus: NoFilu NoFilu NoFilu F p F p p. 5 If h slli hs lsd dys, wh is h pobbiliy h i will ls noh dys? h h Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 of 8 C 8

12 mpl p. 9: Componn filu # A slli hs ling w ub (W) h is usd fo nsmiing micow signl bc o h. h W hs mn-im-o-filu (F) of fou ys. Wh is h pobbiliy h h W lss long hn ys? Lss y o lss? nd h pobbiliy dnsiy funcion (pdf) is f p,, fo fo h pobbiliy disibuion funcion (DF) cn b did s: F p, fo, fo () obbiliy of lsing o mo ys: (b) obbiliy of lsing o lss ys: F p p. 8 F p. (c) obbiliy h i will fil bwn ys nd : F F p p p p.5 F F. 8 ypiclly slli would h mulipl, dundn Ws. hn, if on fild noh would b nbld unil ll ilbl Ws fild. If, fo mpl, w hd Ws, w would b insd in h h filu n nd my no c bou h fis filu ns (h slli sill ops). Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 of 8 C 8

13 h lng Disibuion: h ndom ibl h dscibd h im bwn ny n nd h h following n. hp://n.wiipdi.og/wii/lng_disibuion h lng disibuion ws dlopd by A. K. lng o min h numb of lphon clls which migh b md h sm im o h opos of h swiching sions. his wo on lphon ffic ngining hs bn pndd o consid wiing ims in quuing sysms in gnl. h disibuion is now usd in h fild of sochsic pocsss. h pobbiliy dnsiy funcion (pdf) is f K p,!, fo, fo,,,... h fis momn K K h inc K Wh is h pobbiliy h h fou Ws will ls long hn ys if ch succssi W is no usd unil h pious on fils? f p, fo,!, F p! m Fom bls p d p m fo d m m! m! Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 of 8 C 8

14 Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 of 8 C 8 hfo!! p! F p F p F p F Wh is h pobbiliy h h fou Ws will ls long hn ys if ch succssi W is no usd unil h pious on fils? p F 9. p p F nd finlly F Significnly b hn h pious sul wh w hd 8. p p F Fo ciicl pplicion, dundn dics gully usd. Whn his hppns, filus mus b dcd nd mns o plc h fild uni wih sp mus b ilbl.

15 Disc Dnsiy Funcions Binomil Disibuion h pobbiliy mss funcion (pmf) is f B n n n p p h pobbiliy disibuion funcion (DF) cn b did s: n n FB n p p u hp://n.wiipdi.og/wii/binomil_disibuion Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 5 of 8 C 8

16 Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 of 8 C 8 oisson Disibuion Somims λ is n o b h, i.., h g numb of occuncs p uni im. In h cs, if N is h numb of occuncs bfo im hn w h: h pobbiliy mss funcion (pmf) is,,.!, fo f N o h pobbiliy disibuion funcion (DF) cn b did s: f F!, Alnly, fom h boo,,.! nd fo f nd No h!! f Gin h mh quilnc!! f hn! f!!!

17 Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 7 of 8 C 8 And compuing h scond momn! f!!!!!! hn hp://n.wiipdi.og/wii/oisson_disibuion mpls of ns h cn b modlld s oisson disibuions includ: h numb of unsbl nucli h dcyd wihin gin piod of im in pic of dioci subsnc. h numb of cs h pss hough cin poin on od duing gin piod of im. h numb of splling miss scy ms whil yping singl pg. h numb of phon clls cll cn p minu. h numb of ims wb s is ccssd p minu. Fo insnc, h numb of dis p hou codd on Wiipdi's cn Chngs pg follows n ppoimly oisson disibuion. h numb of odill found p uni lngh of od (his on oo). h numb of muions in gin sch of DNA f cin moun of diion. h numb of pin s p uni of mid fos. h numb of ss in gin olum of spc. h disibuion of isul cpo clls in h in of h humn y.

18 oisson mpl h 5 ypos in boo of 75 pgs. Assum h h ypos p pg oisson Disibud. 5 L X qul h numb of ypos on gin pg. hn os p pg 75 Wh is h pobbiliy h h is on ypo on pg? L = pg, N =X, h numb of ypos on h pg! N f. 5 N f.!! N f. N f. 5! S lb mpl fo numb of os in mulipl pg goups =: o Disibuion Funcion fo pgs =:5 pobbiliy = = = = = numb of os Nos nd figus bsd on o n fom mils in h cous boo: obbilisic hods of Signl nd Sysm Anlysis (d d.) by Gog. Coop nd Cl D. cgillm; Ofod ss, 999. ISBN: B.J. Bzuin, Sping 5 8 of 8 C 8