notes9 - A Lochl'cm—chlz Pam” O70 dx's‘fribu‘ffoms...

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Unformatted text preview: A Lochl'cm—chlz Pam” O70 dx's‘fribu‘ffoms has dans’r‘ries (43067073) OF “HA2. 79mm 3(061/50‘) == 7'; WU—C—g‘é—E‘L) where ExamEle 3 The normal dISTLrIbMJH'OHS are CL Loch—{omnSCQ‘E ‘FCIMJV . “78/2" Tina Shy/dare? ) ——— £ ( -.: I Q, QKQ ’VJW‘) [Vi—fl normal dis‘h’). “a“ 39%“) = 7;: WE") o'fi—TL 2, .. [ €~0¢79/2a'2 ”We” {5 “Hm pdvc we a NQAJO’Z) offs‘fh. EXaleQZ The Cauohx Locq‘h'ohvSCale famfly. TQKQ WW) :: 74—71—721 2 ~00 <PC<OO . ”men 3(7CI/430'): Sty/(92:3) aha 0" is QQi‘fhe Sfavdard devicd’iah. (Same. remark applies In Wex‘f example.) w 3 The, Um'vcorm dishn’s ‘Farm 0L L“3 70am” . _. __ I (JP O<x<| TQKQ 1/4de [(0)000 ~ {O o‘lheerse . V! is ‘Hne Pd-F 0F ‘Hne, UniForm(O)I) dis‘M. Then 3(xl/u’d) .: %EXP(%) WHO)“ ?5 ”WE Pd? 0? We Uniflorm (fig/4+0) dish”). Leif W be 0» 1061?. A Scale Famfly 0F 09 0F “PM Form 3(7Cl0’) = $1 (fig?) where 0”> 0. <0 ‘5 The, SCQ‘C Parame+€r ) T$+n3 has Jensfl‘les A locojion Fama'y 07C OhS‘h’VS has denshLies O‘F We ‘Form 3(%//A)::17V(7L74) where —- 00 </u< ca, Exemplgi The N9“ 1) dis—{735 FOPm CL ) IOCQTFOH Pamfly. Th1 N“); 02) OUSJmS. ‘Farm 6L 9/ Scale. family, “76 Kg WW2) = -L Q 2. Then Ta W __ ,- l \WVCW) " ~13" Q OCH 2 WWW Is‘fine m P0170 070 NM)”; ‘ 576/2”; which {5 ‘HM EXQMEIQ 1 TM, Family 0%: Gamwm (“w/8) charms (where, «O is any 431x96? value we“) Forms CL scale Family. Take \l/(x) : x Q 3 x>o, WW0) I x __ I 0"“)C/m Then prg: __ F6756; Q Woe) O—‘Xo FOR/0) which VS The 13d? 010mm GCIWMCYO, 0”) dish). 3193;: I; we, permfl’ bo‘Pn (3(an l9 +0 Vary) ”We WCQMHX 0F Gammdogffi) offs‘hfis. 0(035 [pi "FOFW'I Q Locq‘Hoh’SCCIIQ fiYMHy (q IS a’fihapé'pamme+€r.) ExEonqvfood Famflias The quHy 05: {ad-F33 or prmcyS {1906(6) 19 669} L The, Parame‘fer Space. (6 miah+ represevfl” a SW15 le Paramder or a. Vecfror 043 paramd’érs.) {S cm ex onen‘fl'd 72am“ 1"]: We can WPHLQ K “flat—hm e "“ . )cC )W(%;,Wm(e)tim)) vafid 1C0)” CL” 76 and all Ge 6). Thfis is +he general K Eqmmder agomnfid 1Cqm1ly (Mama). , For K::\J *Hne Seward Wm] ‘FQYVMW (1P9?) has We, ‘Form Mode) = hm C(69) ©<1O{W(e)t(x)} valid for a” 96 and! 0L” 6669‘ Egg: We aliow h ‘f’o be degenercd‘z (confian‘m buff raga“? 341 ‘Hne OMET funda'om' +0 be nondegenel‘cd’e (woncons+an‘1’) , Emmges mo 1139ch W 4mm) :— i5 5M3) 90 o) ,s>o. (/9415) In ‘Hnis examPle, (0)00) 18 ,8 ° W W N96) C(e) a?) 33;) Thus €(Xlfi) Forms q 1 Q~F W‘HA (Ls idenThCmd above. P l ‘me par'fs B'momiod oflis’rribdfions The ‘Famfly 0F Bihomia‘anp)diS1—ribllhms W‘I‘H'x n Known (10?er) is a Moe-F. The Pm; is Exa 125‘ me 2 210’s. ’.L_ ._. 9&— Wadi/(1)04): I 6 21.4% «oo<x<oo Vl‘ntd' J VCUH For 61> o and? ..Q;p</LK oo ‘ Here, 6 = (/0904) (map @caigujal): Jl>O amp «oo</u<oo} “WW: 4-”- wffi +29; ~z‘f} W0... 20,2 0'9. 20-7— : .L. .1. -— ’- -._L ’- M’fi wwgg’z.) expizofl— X 42%”: 7‘} WV \ A __ , hm) me) w,(e)£3(0<) \/ wate)t1(%) We. have a lpZ‘F WILTIG ‘HOQ ParTs as admfifiefl above. N012: In wri+in om €>100heflfid Hindi/y) h is EWQI +0 be 8€n€m+€ (Cons‘i‘an‘i’): bbd' @‘I any GP‘HM o‘Wer Par'TS. A130) We. Feguire, M96) 20 fora/l 7c. Non~ exeonen‘hm {30mm} I [85 There are, many {Emilia o1" chs‘rr‘lbu‘HOhS which are QQI exPonemLM Famxhes. The Cauchy Location-Scale Famfly fix a” % ._._ I . \ “WW“ 2.:— wows—2&1) [email protected] be Wrifieh as an exponen‘fial fim'sy. (Try H"! ) This is ngj' V4130? 100F391) 7C} bquonIy for %>/&. T6 cowL om apression Valid? 10ch an 96) we new? an indicq‘for Puncf‘fon. Eagr f The SULPPOH' 07C an expon en‘fial ‘FQM\\)’ O‘F 700910)stme 100619) is “I‘he same, fir a.“ 6. Egg? (For 190C)? The suppor‘!’ 610 Note) =—= hm d6) exp; wremw} 3's fad h(oc)>o} which has no‘f ’valVe 6. Rdum +0 Previous Examge, The. p51? ' .( , “cam/93 “ 753 99% For 96% has suppor‘l’ {96276514.} which dame/5 on Gsoujlg) +hroo5>h *Hwa value/u, Thus (wi‘Hnod Fur‘ffier work) wz know fhis is ad“ an exPohenfiai Family- [M3 The, Fam'fly 010 Uni¥orm(3_)b) omhxmfions w'er ~oo< a<b < 00 55 nit an exponefi‘fial ‘Famfly. "WW— UmigormmJb) alansfly Roclajb) = 31:35 1Cor a<x<b W O o‘H'reerQ Wé KWW ””115 35 [191’ an exponern‘fql 100ml)». W 37916 Cauchy Iocq+fm_gch 1%”!in {S 053 01‘ exponevrfiql Family) but {1-5 Suppofl' {SWW ‘FOraH 9:9“)0'). ...
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