sthiramaaya enna vitharanam

probability sidhaanthathil, sthiramaaya kaundu distribution ennathu ekapaksheeyamaaya sthiramaaya vitharanathinu munbulla konjugettaanu. ezz & pii 500, wix ennivayude dainamdina vitharanangalekkurichulla 2017 leeii padanatthil steefan lihn (chinesa <unk>) eee vitharanam kandethi. sthirathayulla vitharana kudumbatthe chilappol levi alfa-sthiramaaya vitharanam ennum vilikkunnu, ithu padicha aadyathe ganithashaasthranjanaaya pooll levic sesham.
vitharanathe nirvachikkunna moonnu paaraamettarukalil, sthiratha paaraameettar formula _ 1 ettavum pradhaanamaan. sthiramaaya kaundu vitharanagalkk formula 2 undu. formula 3-nte ariyappedunna vishakalana cases wix vitharanavumayi bandhappettirikkunnu (sekshan 7 kaanuka). vitharanathinu ellah nimishangalum parimithamaanu.
nirvachanam.
athinte staanderd distribution nirvachichirikkunnathu ingane
formula _ 5 umm formula _ 6 umm evideyaanu
athinte lokkeshan skeyil kudumbatthe ingane nirvachichirikkunnu
formula _ 8, formula _ 9, formula _ 6 enniva
melpparanja padaprayogatthil, formula _ 11 oru ekapaksheeyamaaya sthiramaaya vitharanamaanu, athu inipparayunna reethiyil nirvachikkappedunnu.
formula _ 12 oru staanderd stebal random veriyabilaayirikkatte, athinte vitharanam formula _ 13 kondu savisheshamaan, appol namukk undu
evide soothravaakyam _ 2.
levi summ formula 16 pariganikkuka, avide formula 17, formula 18 nuu density formula 19 undu, avide formula 20. formula 21 sajjamaakkuka, normalisation sthiraankamillaathe nammal formula 22 ll athunnu.
eee vitharanathe "sthiramaaya ennam" ennu vilikkunnathinulla kaaranam 23 enna bandha soothravaakyam kondu manassilakkaan kazhiyum. 24 enna soothravaakyam levi thukayude "ennam" aanennu shradhikkuka. oru nischitha soothravaakyam nalkiyaal, eee vitharanam oru yoonit dooram sanjarikkunnathinu 24 gattangal enna soothravaakyam edukkaanulla saadhyatha nalkunnu.
avibhaajya roopam.
formula _ 11, formula _ 28 ennivayude avibhaajya roopathe adisthaanamaakki, namukk formula _ 29 nte avibhaajya roopamundu.
mukalilulla eratta-saine integraline adisthaanamaakki, ithu staanderd sii. dii. ephinte avibhaajya roopathilekku nayikkunnu
ivide soothravaakyam _ 32 ennathu saine integral functionaanu.
sariyaaya praathinidhyam.
seerees representationil", sthiramaaya kaundu distribution raitt functionte oru prathyeka kesaanennu kaanikkunnu (sekshan 4 kaanuka):
ithu hankel integralilekku nayikkunnu (1.4.3 adisthaanamaakki)
badal vyulppannam-lamda vighadanam.
sthiramaaya kaundu distribution nedunnathinulla mattoru sameepanam ekapaksheeyamaaya sthiramaaya vitharanathinte laplace parivartthanam upayogikkuka ennathaanu, (sekshan 2.4)
formula _ 37, oraalkk idathu vashathulla integraline oru staanderd laplace vitharanathinteyum staanderd stebal kaundu vitharanathinteyum uthppanna vitharanamaayi vighadippikkaan kazhiyum,
evide formula _ 39.
lihnte muun kruthikalil elhsinu "cemetric lamda distribution" ennu perittathinaal ithine "lamda decomposition" ennu vilikkunnu (sekshan 4 kaanuka). ennirunnaalum, ithinu "exponential power distribution" allengil "generalised error/normal distribution" polulla niravadhi janapriya perukalundu, ithu formula _ 40 aayirikkumbol palappozhum paraamarshikkappedunnu. ithu reliability enchineeyaringile weibul survival function koodiyaanu.
sthiramaaya niyamathinu keezhilulla lihnte asattu ritten chattakkoodinte aditharayaanu lamda vighadanam. aasthi rittenukalude vitharanamaanu elhs. rhs-ll, laplace vitharanam lepcurtotic shabdathe prathinidheekarikkunnu, sthiramaaya kaundu vitharanam asthirathaye prathinidheekarikkunnu.
sthiramaaya wolt vitharanam.
sthiramaaya kaundu vitharanathinte oru vakabhedathe sthiramaaya wolt vitharana soothravaakyam _ 41 ennu vilikkunnu. veriyabilinte maattatthiloode lamda vighadanathil ninnu ithu uruthirinjekkaam (sekshan 6 kaanuka). formula _ 42-nte laplace parivartthanam oru gausiyan mishrithatthinte adisthaanathil prakadippikkunnu,
evide
eee parivarthanatthinu saamaanyavalkkaricha gass parivartthanam ennu vilikkunnu, kaaranam ithu formula _ 45 nuu thulyamaaya gass-laplace parivartthanatthe saamaanyavalkkarikkunnu.
gaama, poyson vitharanangalumayulla bandham.
gaama, poyson vitharanagalude aakruthi paaraameettar leviyude sthiratha paaraameettar formula _ 46-nte vipareethavumaayi bandhippichirikkunnu. 
mukalile kramappeduthiya gaama function formula _ 47 formula _ 48-nte apoornnamaaya avibhaajyaghadakamaayi prakadippikkaam 
formula _ 49
formula _ 48-neey vighadanam upayogichu maattisthaapikkukayum oru avibhaajyaghadakam nadappilaakkukayum cheiyunnathiloode namukk undu
soothravaakyam _ 51
formula _ 52 formula _ 53 lake thirichuvannaal, sthiramaaya ennathinte adisthaanathil formula _ 47-nte vighadanathilekku nammal athunnu 
soothravaakyam _ 55
formula _ 47-neey formula _ 57 kondu verthirikkuka, nammal aavasyamulla formula athunnu
ithu oru uthppanna vitharanathinte roopathilaanu. rhs-leeii formula 59 enna padam aakruthi formula 60-nte oru weibul vitharanavumayi bandhappettirikkunnu. athinaal, eee formula sthiramaaya kaundu vitharanathe oru gaama vitharanathinte probability density functionumaayi (ivide) bandhippikkunnu. koodaathe shape paaraameettar formula _ 60 leviyude stebility paaraameettar formula _ 46-nte vipareethamaayi kanakkakkaam.
chii, chii-squared vitharanangalumayulla bandham.
chii, chii-squared vitharanangalile digri oof freedam formula 64 formula 65 maayi bandhappettirikkunnave kaanikkam. athinaal, lamda vighadanathil formula 66 oru poornnasankhya soochikayaayi kaanuka enna yadhaartha ashayam ivide nyaayeekarikkunnu.
chii-square vitharanathinu, chii-square vitharanam gaama vitharanathinte oru prathyeka kesayatinal, aaa formula 67. mukalil ninnu, oru gaama vitharanathinte aakruthi paaraameettar formula 46 aanu.
chii vitharanathinayi, nammal athinte cdf formula _ 69 ll ninnu aarambikkunnu, avide formula _ 70. formula _ 69 neey formula _ 57 kondu verthirikkuka, namukk athinte saandrata pravartthanam undu
eee soothravaakyam _ 74 enna soothravaakyathe _ 76 padatthiloode _ 1 enna soothravaakyavumaayi bandhippikkunnu.
pothuvalkkaricha gaama vitharanangalumayulla bandham.
pothuvalkkaricha gaama vitharanam rund aakruthiyilulla paaraamettarukalulla oru probability vitharanamaanu, ithu gaama vitharanathinte suuppar settu aanu, weibul vitharanam, exponential vitharanam, pakuthi-saadhaarana vitharanam. athinte sii. dii. eph. formula _ 77 roopathilaanu.
formula _ 81-neey formula _ 57 kondu verthirikkuka, nammal uthppanna-vitharana formula athunnu
ivide formula _ 84 oru saamaanyavalkkaricha gaama vitharanathinte pdf soochippikkunnu, 
ithinte sii. dii. eph. formula _ 81 aayi paaraameettar cheythirikkunnu. 
eee soothravaakyam formula _ 86 neey formula _ 1 maayi formula _ 88 padatthiloode bandhippikkunnu. formula _ 89 padam aakruthi-paaraameettar sthalathe second digri oof freedam prathinidheekarikkunna oru exponentaanu.
asymptotic gunangal.
sthirathayulla vitharana kudumbatthinu, athinte lakshanangalillaatha perumattangal manassilaakkendathu athyaavasyamaanu. muthal, cheria soothravaakyam _ 90 nuu,
ithu formula _ 92 sthireekarikkunnu.
valia soothravaakyam _ 90 nuu,
95 enna soothravaakyathinte wall ananthathayil ganyamaayi shayikkunnuvennu ithu kaanikkunnu. 96 enna valia soothravaakyam, kshayathe kooduthal shakthamaakkunnu.
eee wall oru saamaanyavalkkaricha gamma vitharanathinte roopathilaanu, athinte soothravaakyathil _ 97 paaraameettartaiseshan,
formula _ 98, formula _ 99, formula _ 100. 
athinaal, ithu formula _ 101 nuu thulyamaanu, 
ithinte sii. dii. eph. formula _ 102 aayi paaraameettar cheythirikkunnu.
nimishangal.
22 enna soothravaakyathinte "n"-th moment formula 103 ennathu 11 enna soothravaakyathinte 105th moment aanu. ellah positteev momentukalum parimithamaanu. sthiramaaya vitharanathil vyathichalikkunna nimishangalude mulpadarnna prashnam ithu oru tharatthil pariharikkunnu. (bhagam 2.4 kaanuka)
nimishangalude vishakalana pariharam raitt functioniloode labikkum
evide formula _ 109 ((1.4.28 kaanuka))
athinaal, soothravaakyathinte sharashari _ 22 aanu
vyathyasam ithaanu
ettavum kuranja moment formula _ 113 aanu.
moment generating function.
oru phoox-raitt function allengil phoox echu-function vazhi mgf prakadippikkaam
oru parisodhanayaayi, formula _ 115-ll, formula _ 116 (chuvade kaanuka) formula _ 118 vazhi formula _ 117 aayi vikasippikkaam.
ariyappedunna vishakalana cases-quartic sthirathayulla ennam.
formula 119 aayirikkumbol, formula 120 ennathu oru vipareeta gaama vitharanamaaya levi vitharanamaanu. athinaal formula _ 121 ennathu 3/2 aakruthiyudeyum skeyil formula _ 122-nteyum oru shift cheytha gaama vitharanamaanu.
ivide formula _ 8, formula _ 9.
athinte sharashari formula _ 126 umm athinte staanderd deviation formula _ 127 umm aanu. ithine "quartic steble kaundu distribution" ennu vilikkunnu. quartic" enna vaakk lamda vitharanathekurichulla linte muun kruthiyil ninnaan vannath, avide formula _ 128. eee krameekaranatthil, sthiramaaya kaundu vitharanathinte pala vashangalkkum gambheeramaaya vishakalana parihaarangalund.
p"-th kendra nimishangal formula _ 129 aanu. cdf formula _ 130 aanu, ivide formula _ 131 ennathu apoornnamaaya gaama functionaanu. mgf formula _ 132 aanu. (bhagam 3 kaanuka)
α → 1 aayirikkumbol prathyeka cases.
formula _ 133 valuthaakumbol, vitharanathinte peak moorchayullathaayitheerunna. formula _ 22-nte oru prathyeka cases formula _ 135. vitharanam oru diarac deltta function pole pravartthikkunnu,
ivide formula _ 137, formula _ 138 enniva undu.
parambara praathinidhyam.
ekapaksheeyamaaya sthiramaaya vitharanathinte parambara praathinidhyathe adisthaanamaakki, namukk undu
eee parambarayude praathinidhyathinu rund vyaakhyaanangalundu
gamma functionte reflection formula vazhiyaan thelivu labhikkunnatha formula 145, ithu mapping amgeekarikkunnu formula 143 leeii formula _ 145. sariyaaya praathinidhyam sthiramaaya kaundu vitharanathinte niravadhi stattistikkal propertykalkku vishakalana parihaarangalilekku nayikkukayum fraakshanal calculusumaayi mattoru bandham sthaapikkukayum cheiyunnu.
aaplikkeshanukal.
sthiramaaya kaundu vitharanam viksinte dainamdina vitharanathe nannayi prathinideekarikkum. wix formula _ 148 pole formula _ 149, formula _ 150 enniva upayogichu vitharanam cheyyappedunnuvennu anumaanikkappedunnu (sekshan 7 kaanuka). athinaal sthiramaaya kaundu distribution ennathu oru asthiratha prakriyayude aadya-oordar marginal distributionaanu. eee saahacharyathil, formula _ 151 neey "floor volattilitti" ennu vilikkunnu. praayogikamaayi, wix apoorvvamaayi 10-ll thaazheyaan. eee prathibhasam "floor volattilitti" enna ashayathe nyaayeekarikkunnu. fittinte oru sample chuvade kaanichirikkunnu 
formula _ 152-nulla sharashari-reverting ezz. dii. iii. yude oru roopam oru parishkaricha cox-singersol-rooss (sii. aii. aarr) modaline adisthaanamaakkiyullathaanu. formula _ 153 asthiratha prakriyayaanennu karuthuka, namukk undu
ivide soothravaakyam _ 155 ennathu "wall oof wall" ennu vilikkappedunnu. viksinte "wall oof wall" neey vivix ennu vilikkunnu, ithinu ekadesham 85 enna saadhaarana moolyamundu.
eee ezz. dii. iii vishakalanaparamayi tractable aanu, koodaathe feller avasthaye thripthippeduthunnu, athinaal formula 153 orikkalum formula _ 151 nuu thazheyaakilla. ennaal sidhaanthavum prayogavum thammil oru sookshumamaaya prashnamund. wix formula _ 151 nuu thaazheyaayathin ekadesham 0.6% saadhyathayund. ithine "spillovar" ennu vilikkunnu. ithine abhisambodhana cheiyunnathinu, oraalkk square root padam formula 159 upayogichu maattisthaapikkaam, avide formula _ 160 formula _ 153 nuu formula _ 151 nuu alpam thazheyaayi neengaan oru cheria chorcha chaanal nalkunnu.
valare kuranja wix reeding valare samthrupthamaaya oru vpaniye soochippikkunnu. athinaal spillovar avastha, formula _ 163, oru prathyeka praadhaanyam vahikkunnu-athu sambhavikkumbol, ithu saadhaaranayaayi bisinesses sykkilile kodunkaattinu munbulla shaanthathaye soochippikkunnu.
kramarahithamaya veriyabilukalude uthpaadanam.
mukalil kaanichirikkunna parishkkaricha sir model kaanikkunnathupole, sthiramaaya kaundu random veriyabilukalude sequenceukal anukarikkaan mattoru inputt paaraameettar formula _ 164 aavasyamaanu. sharashari-reverting stokkaastik prakriya roopamedukkunnu
ithu formula _ 166 undaakkanam, athu formula _ 168 aayi formula _ 167 pole vitharanam cheiyunnu. 
formula _ 170 ethra vegathil maaranam ennathinulla upayokthaav vyakthamakkiya mungananayaan formula _ 164. 
focker-plank samavaakyam pariharikkunnathiloode, soothravaakyam _ 172 nte adisthaanathil _ 171 enna soothravaakyathinte pariharam ithaanu
rund raitt functionukalude anupathamaayum ithu ezhuthaam, 
formula _ 175 aayirikkumbol, eee prakriya parishkaricha sir modalilekku churukkunnu.
soothravaakyam _ 176. 
soothravaakyam _ 177 oru nerrekhayaayirikkunna oreyoru prathyeka cases ithaanu.
sii. aarr. modalinte sthiramaaya vipuleekaranam.
178 enna padavum mukalilulla 164 enna padavum thammilulla karkkashamaaya bandhatthil elavu varuthunnathiloode,
sii. aarr. modalinte sthiramaaya vipuleekaranam nirmmikkaan kazhiyum
ithu yadhaartha sir modalilekku churukki
formula _ 175 lham 
formula _ 182. 
athinaal, paaraameettar formula _ 79 sharashari-reverting vegathaye niyanthrikkunnu, 
lokkeshan paaraameettar formula _ 184 settukal sharashari evideyaano, formula _ 164 asthiratha paaraameteraanu, 
sthiramaaya niyamatthinte aakruthi paaraameteraanu formula _ 1.
focker-plank samavaakyam pariharikkunnathiloode, formula _ 188-leeii pdf formula _ 187-nte pariharam ithaanu
eee pariharam manassilakkaan, valia soothravaakyam _ 57-nuu lakshanangalillaatha reethiyil pariganikkuka, formula _ 187-nte wall ippozhum oru saamaanyavalkkaricha gamma vitharanathinte roopathilaanu, athinte soothravaakyam _ 192 paaraameettariseshanil,
formula _ 98, 
formula _ 194, oppam 
formula _ 195. 
ithu yadhaartha sir modalilekku churukkiyirikkunnu
formula _ 175 ll formula _ 197 
formula _ 198 oppam 
formula _ 199; athinaal
formula _ 200.
fraakshanal calculus.
mittag-lefler functionumaayi bandhappettirikkunnu.
sekshan 4 ll ninnu, mittag-lefler function formula _ 202-nte vipareeta laplace parivartthana formula _ 201 aanu (formula _ 203)
maruvashathu, pollard (1948) thaazhepparayunna bandham nalki.
athinaal _ 206 enna soothravaakyathiloode, sthiramaaya kaundu vitharanavum mittag-lephttar function thammilulla bandham namukk labikkum
formula _ 209, formula _ 210 ennivayil eee bandham vegathil parisodhikkaan kazhiyum. ithu ariyappedunna quartic sthirathayulla kaundu falatthilekku nayikkunnu
time-fraakshanal focker-plank samavakyavumaayulla bandham.
saadhaarana focker-plank samavaakyam (fpe) formula _ 212 aanu, ivide formula _ 233 focker-plank space oopparettaraanu, formula _ 214 defusion coffificient aanu, formula _ 215 thapanilayum formula _ 216 baahya feeldumaanu. time-fraakshanal efpe adhika fraakshanal derivative formula _ 217 avatharippikkunnu, athaayath formula _ 218, ivide formula _ 219 fraakshanal defusion coffificient aanu.
formula _ 201 ll formula _ 220 aanenkil, time-fraakshanal fpe (eq (16))-nulla kernel namukk labikkum.
ithil ninnu oru saadhaarana parihara soothravaakyamaaya _ 224 ll ninnu bhinna saandrata soothravaakyam _ 223 kanakkakkaam.
variable formula _ 227-nte maattatthiloode formula _ 226 aayathinal, mukalil paranja integral "lamda decomposition" ashayathinum time formula _ 229-nte scalingin samanamaaya formula _ 228 ulla uthppanna vitharanamaayi maarunnu
ivide soothravaakyam _ 232 ennathu asamathva vyaapanathin kaaranamakunna _ 232 enna soothravaakyathinte yoonittil prakadippikkunna ashudhiyude vitharanamaayi vyaakhyaanikkappedunnu.
weibul vitharanavumayi bandhappettathaan.
cdf formula _ 233 aaya oru weibul vitharanathinu, athinte aakruthi paaraameettar formula _ 234 leviyude sthiratha paaraameettar formula _ 235 nuu thulyamaanu. 
uthppanna vitharanathinte samanamaaya oru aavishkaaram uruthirinjekkaam, athaayath kernel onnukil 
oru ekapaksheeya laplace vitharana formula _ 236 allengil oru railey vitharana formula _ 237. 
lamda vighadanathil ninnu varunna pooraka sii. dii. ephil ninnaan ithu aarambhikkunnath
formula _ 57 ll derivative edukkunnathiloode, oru weibul distribution pdf formula _ 240-nte uthppanna vitharana roopam namukk labikkum.
ivide formula _ 242 umm formula _ 243 umm undu.
245 enna soothravaakyathil ninnum 246 enna soothravaakyathil ninnum 244 enna soothravaakyam vyakthamaan.

