dishatmak aakadevari

dishatmak sankhyiki (golakar sankhyiki kinwa golakar sankhyiki dekhil) hii sankhyikichi upashakha aahe jee disha (euclidean spacemadhil ekak vahak, r "n"), aksh (r "n" madheel mool resha) kinwa r "n" madheel paribhramananshi sambandhit aahe. adhik samanyatah, dishatmak aakadevari hii stephel manifolds sankshipt remanion manifoldsvaril nirikshananshi sambandhit aahe.
0 ansh aanee 360 ansh heey samaan coone aahet, jenekarun udaharanarth 180 ansh heey 2 ansh aanee 358 anshanche samajanyasarkhe sarasari nahee, heey eka udaharan dete qee kahee prakarchya detachya vishleshanasathi vishesh sankhyikeey paddhati aavashyak aahet (yaa prakaranat, koneey deta). nirdeshatmak manalya bhagatoon mahitichya itar udaharananmadhye tatpuratya kalavadhishi sambandhit aakadevaricha samavesh hoto (udaa.jee. divas, aathvada, mahina, varsh ii. chaa kalavadhi.), kampas disha, renunmadhil dihedral coone, disha, paribhraman ityadi.
paripatrak vitaran.
reshevaril konatehi sambhavyata ghanata karya (pii. dii. eff.) sootra _ 1 heey ekak trijya asalelya vartulachya parighajaval 'gundalale' jaau shakate. mhanje, gundallelya chalchi pii. dii. eff. 
sootra _ 2
aahe
sootra _ 3
hii sankalpana bahubhinnatechya sandarbhat, vaishishtya jaaget serve parimanancha samavesh asalelya anek sutranchya 4 berijparyant sopya bereejcha vistar karoon vistarli jaau shakateh
sootra _ 5
jethe sootra _ 6 heey sootra _ 7 weey euclidean besis vector aahe.
khaleel vibhag kahee sambandhit paripatrak vitaran darshavtat.
wonn misesache golakar vitaran.
wonn misses vitaran" heey eka golakar vitaran aahe, jeey itar konatyahi golakar vitaranapramanech, vartulachya bhovati vishisht rekheey sambhavyata vitaranache aavaran mhanoon manale jaau shakate. wonn misses vitaranasathiche antarnihit resheey sambhavyata vitaran ganitiyadrushtya avyavharya aahe; tathapi, sankhyikeey hetunsathi, antarnihit resheey vitaran hatalanyachi aavashyakta nahee. wonn misses vitaranachi upayuktata duppat aaheh serve golakar vitarananpaiki heey sarvat ganitiyadrushtya shodhanyayogya aahe, jyamule sope sankhyikeey vishleshan shakya hote aanee heey gundallelya samanya vitaranachya jawalche andaje aahe, jeey rekheey samanya vitaranashi samaan aahe, kaaran heey mothya sankhyene lahan koneey vichalananchya berijsathi maryadit prakaran aahe. kharey tarr, wonn misesache vitaran anekada 'golakar samanya' vitaran mhanoon olakhale jaate kaaran tyacha vapar sulabh aahe aanee tyacha gundallelya samanya vitaranashi jawalcha sambandh aahe (fisher, 1993).
wonn misses vitaranachi pii. dii. eff. aaheh sootra _ 8 jethe sootra _ 9 heey 0 kramache sudharit besal karya aahe.
eksaman vartulakar vitaran.
golakar eksaman vitaran" chey sambhavyata ghanata karya (pii. dii. eff.) khalilpramane dile aaheh
sootra _ 10
vareel wonn misesache sootra _ 11 mhanoon dekhil yacha vichar keela jaau shakato.
samanya vitaran gundallele.
rapped normal distribution' (doubleu. aine.) chee pii. dii. eff. ashee aaheh
sootra _ 12
jethe anukrame, naan gundallelya vitaranache sarasari aanee manak vichalan heey, micro aanee σ aahet aanee sootra _ 13 heey jakobi theeta karya aaheh
sootra _ 14 jethe sootra _ 15 aanee sootra _ 16
gundallele kachee vitaran.
rapped kauchi distribution' (doubleu. see.) chee pii. dii. eff. ashee aaheh
sootra _ 17
jethe sootra _ 18 haah scale ghatak aahe aanee sootra _ 19 heey shikhar sthan aahe.
gundallelya leviche vitaran.
rapped levy distribution' (doubleu. ell.) chee pii. dii. eff. ashee aaheh
sootra _ 20
jethe sootra _ 21 asatana bereejche moolya shoonya manale jaate, sootra _ 22 haah scale ghatak aahe aanee sootra _ 23 haah sthan mapdand aahe.
uchch-aayami anekpatanvar vitaran.
dvimiteey goal (jase qee kent vitaran), "aine"-aayami goal (wonn misses-phisha vitaran) kinwa torrus (bivariate wonn misses vitaran) varr dekhil vitaran astitvat aahe.
matricks wonn misses-fishers vitaran heey stephel manifoldvaril vitaran aahe aanee rotation matrixver sambhavyata vitaran tayar karnyasathi vaparale jaau shakate.
bingham vitaran heey 'aine' parimananmadhil akshanvar kinwa samatulya, ('aine'-1)-aayami golavareel bindunvar, olakhalya gelelya pratidhruvansaha, vitaran aahe. udaharanarth, jarr "n" = 2 asel, tarr aksh heey pratalatil mool bhagatoon bhagatoon dishaheen resha aahet. yaa prakaranat, pratyek aksh heey samatalatil ekak vartul (jeey eka-aayami goal aahe) ekmekanche pratidhruv asalelya done bindunvar kapato. n" = 4 sathee, bingham vitaran heey ekak chaturthanshanchya (versors) jaaget vitaran aahe. versor haah rotation matrixshee sambandhit asalyane, matricks-vonn misses-fishers distributionpramanech, "aine" = 4 sathee bingham distribution vapar rotationchya jaaget sambhavyata vitaran tayar karnyasathi keela jaau shakato.
udaharanarth, heey vitaran bhoogarbhshastra, sfatikalekhan aanee jaiv-mahitishastrat vaparale jaate.
kshan.
golakar vitaranachya kachchya vahak (kinwa trikonamitiy) kshananchi vyakhya ashee kelee jaateh
jethe sootra _ 25 heey lambi sootra _ 26 chey konatehi antar aahe, sootra _ 27 heey golakar vitaranache pii. dii. eff. aahe aanee sootra _ 28. kaaran avibhajya sootra _ 27 heey ekatva aahe aanee ektrikaran antar maryadit aahe, tyamule konatyahi golakar vitaranache kshan nehmich maryadit aanee changlya prakare paribhashit kele jaatat.
namuna kshan samaan ritine paribhashit kele jaatatah
loksankhyecha parinami vahak, lambi aanee sarasari coone sambandhit namuna mapdandanchya sadrushyat paribhashit kele jaatat.
yavyatirikt, uchch kshananchi lambi ashee paribhashit kelee jaateh
uchch kshananche koneey bhag fakt sootra _ 35 aahet. serve kshananchi lambi 0 aanee 1 chya darmyan asel.
sthan aanee prasarache upay.
kendriya pravrutti aanee sankhyikeey prasaran yaa donhi upayanchi vyakhya loksankhya aanee tyaa loksankhyetun ghetalelya namunyasathi kelee jaau shakate. 
madhyavarti pravrutti.
sthanache sarvat samanya mojmap mhanje golakar sarasari. loksankhyecha golakar sarasari haah fakt vitaranacha pahila kshan asato tarr namuna sarasari haah namunyacha pahila kshan asato. namuna sarasari loksankhyechya sarasaricha nihpakshapati andaj mhanoon kaam karel.
jevha deta kendrit asato, tevha madhyika aanee mode resheey prakaranachya sadrushyane paribhashit kele jaau shakatat, parantu adhik vikhuralelya kinwa bahu-model detasathi, yaa sankalpana upayukt nasatat.
prasar.
golakar prasarache sarvat samanya upay pudhilpramane aaheth
madhyamache vitaran.
aine' mojamapachya sutracha sanch dilelya _ 44 chya 'zed' chey sarasari moolya asey paribhashit kele aaheh
jyapramane vyakt kele jaau shakate
kuthe?
kinwa, paryayitah
kuthe?
golakar pdf "p" ("θ") sathee sarasari konache (sootra _ 50) vitaran khalilpramane dile jailah
jethe sootra _ 25 heey sootra _ 26 chya lambichya konatyahi antaravar aahe aanee avibhajya heey nirbandhanchya adheen aahe qee sootra _ 54 aanee sootra _ 55 sthir aahet, kinwa, vaikalpikaritya, teey sootra _ 56 aanee sootra _ 50 sthir aahet.
bahutek golakar vitaranansathi madhya vitaranachi ganana vishleshanatmakdrushtya shakya nahee aanee bhinnateche vishleshan karnyasathi, sankhyatmak kinwa ganiteey andaj aavashyak aahet.
kendriya maryada pramey namuna sadhananchya vitarnavar lagoo kele jaau shakate. (mukhya lekhh dishatmak sankhyikisathi kendriya maryada pramey). heey darshavile jaau shakate qee sootra _ 58 chey vitaran mothya namuna aakarachya maryadet dwibhinna samanya vitaranakade jaate.
tandurustichi gunavatta aanee mahatva chachani.
chakriy mahitisathi-(udaa.jee., teey eksamanpane vitarit kele jaate qaa):

