# On Gaussian Distribution

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1 Prpr b Çğt C MTU ltril gi. Dpt. 30 Sprig 0089 oumt vrio. Gui itributio i i ollow O Gui Ditributio π Th utio i lrl poitiv vlu. Bor llig thi utio probbilit it utio w houl h whthr th r ur th urv i qul to or ot. R. Ar ur Gui Ditributio [Sigl Ali Ppouli] Lt I I th b impl hg o vribl I ; th 0 π π π r rrφ Thi how tht i vli probbilit it utio. R. M Vri Clultio or Gui Ditributio; M It b ot tht th itributio i mmtri rou tht i. W ow tht th m vlu houl b lot t th tr o mmtr th. W how thi rult ollow B impl hg o vribl w gt. But i w hv

2 Prpr b Çğt C MTU ltril gi. Dpt. 30 Sprig 0089 oumt vrio. th. Vri Thi i mor tri [Probbilit Ppouli]. Th r ur th Gui itributio i qul to tht i. ow tig rivtiv o thi π rltio wrt to w gt π π 0 0 Th lt qutio b writt 0 howig o mor tim tht th m o Gui itributio i. Tig rivtiv o 0 rltio with rpt to or o tim w gt 0 howig tht th vri o Gui itributio i. Tig rivtiv wrt to ott uh i hr i powrul lultioproo tool. Thi pro b o i th rltio who rivtiv i t i vli or otiuum o vlu. I our th r ur Gui itributio i or vlu o. H I i vli or. π A o ot i o th itrhg o rivtiv itgrl oprtor. Thi tp loo iot but it i i trhrou. W hv to vri th itrhg i grl. I ou hv o i or oubl i itgrl O i itgrl h uppr or lowr limit ± oubl i itgrl hv upprlowr limit.; th itgrl m or m ot ovrg. Th itrhg o itgrl or rivtiv oprtor i th itrhg o two limitig oprtio. Rmmbr rivtiv i lo i t limit o h 0 or h. O h to b rul bout thi itrhg. Th itgrl tui i th probbilit our i grl bhv il poitiv uit r utio. But w

3 Prpr b Çğt C MTU ltril gi. Dpt. 30 Sprig 0089 oumt vrio. houl lw b rul wh w hg oprtor ivolvig iiit or iiitiml qutiti. R.3 Th itributio o b whr i orml itribut Th itributio o b b pr uig th umtl thorm or b utio o rom vribl. Wh th iitio or Gui it i ubtitut or w gt b π From th lt qutio w ot tht b vri b. π. i lo Gui itribut with m W rh importt oluio tht b lig biig Gui r.v. w gt othr Gui r.v. with irt m vri. Th m vri o th w Gui rom vribl b il lult uig pttio oprtor b b imilr rltio or vri. A th itributio o b writt b uig th lult m vri. W ow tht i rom vribl i Gui itribut w ol to lult it m vri to writ th itributio. A mpl lt 0 Gui itribut or orml itribut with ro m uit vri th w 0 0 i -000 i th m o 0 0 i -0 it vri i w w R.4 Momt Grtig Futio Lt b ro m uit vri Gui itributio 0. Th momt grtig utio o i. W writ - π prt qur orrtio trm ollow. Thi pro i ow ompltig th qur rqutl u. Wh thi rltio i ubtitut to w gt

4 Prpr b Çğt C MTU ltril gi. Dpt. 30 Sprig 0089 oumt vrio. π π π. Th momt grtig utio i vli or ll vlu. Wh i Gui rom vribl with m vri ; th w ow tht b writt whr. 0 Th momt grtig utio o i qul to. R.5 Highr Orr Momt o Gui Ditributio Lt w ow tht. Uig powr ri pio o whih i 0! w writ ollow Th oiit o i i qul to!. Wh 0 w hv !! m R.6 Aitio o two ipt Gui r.v. i Gui r.v. Wh two Gui itribut ipt rom vribl r th rultt itributio i lo Gui itribut whr.. Thi i proprt o grt import i prti. Thi rult b il how uig momt grtig utio. whr w hv u th ip o i th lt qulit. From R.4 w ubtitut th momt grtig utio o Gui itributio gt

5 Prpr b Çğt C MTU ltril gi. Dpt. 30 Sprig 0089 oumt vrio. w olu tht vri.. Comprig thi rult with R.4 i i Gui itribut with m I th t tio w i joit Guiit how tht rbitrr lir ombitio o Gui rom vribl rult i Gui itributio. Joitl Gui Rom Vribl W irt mi two rom vribl th t to rom vtor whih i th joit itributio o rom vribl. Two rom vribl writt i th ollowig orm r ll joitl Gui i thir joit it b p π Th utio i lrl poitiv vlu o iit r i uppr bou b - or lrg thror with propr lig it b utili it utio. ot tht i utio o two ipt vribl h iv prmtr. W will tth th lbl o m vri orrltio oiit to th vribl tr jutiig wh w o o. A ltrtiv but quivlt iitio or joit Guiit b giv ollow r ll joitl Gui i ϖ ϖ i Gui itribut or vlu o ϖ ϖ. Hr i th tr uivribl Gui itributio tht w hv prvioul mi. lo ll uivrit Gui W t grt th quivl o two iitio pro without proo or ow. Th utio g rlt riptio giv i R.4 giv blow b ormult proo.

6 Prpr b Çğt C MTU ltril gi. Dpt. 30 Sprig 0089 oumt vrio. R. Diitio i mtri orm Th iitio lo b writt ollow [ ] p π ot tht th rgumt o th potil mimi th o imio Gui itributio. I ou tr ot to th trm rlt to vribl th ou hv [ ] p p From hr w l tht multi-imiol Gui itributio i li to uivrit igl vribl Gui itributio whih i t who til r mi blow. Th quivl o mtri iitio giv i R. to th origil iitio ollow vr impl b lultig th mtri ivr i th mtri iitio R. Diitio or ro m joitl itribut Gui r.v. Wh ro i ubtitut or th m w gt th ollowig [ ] p π I w rom vribl r i rom whr r joitl Gui r.v. with ro m th joit p i giv bov uh tht Th itributio o w rom vribl i b umtl thorm w tht th itributio o h th orm o grl joit Gui it. Si m vlu ol hit th tr o mmtr to poit o pl omtim w prr to wor with ro m Gui itributio

7 Prpr b Çğt C MTU ltril gi. Dpt. 30 Sprig 0089 oumt vrio. tht thi rult i lo vli or o-ro m Gui itributio. Thi i u to th mtio hit i th tr o mmtr. W hv om mpl o thi itutio i th ot. R.3 Two ipt Gui r.v. r joitl Gui Lt r ipt Gui itribut ro m rom vribl. A ot i R. th vlu o th m i ot importt i thi rgumt. W prr to t th m vlu ro to impl th prttio. Th wh th iitio or th uivrit Gui itributio i irt w gt th iitio o joit Gui itributio or ro m vribl. Si th joit it o two ipt Gui rom vribl b writt i th orm rquir b th th joit itributio o Gui iti two ipt Gui rom vribl r i joitl Gui. Thi i ot vr urpriig th lbl o th itributio giv w th il oluio. A mor urpriig rult i th t o. I but ot ipt th joit itributio o Tht i i mrgil iti o ril Gui. A impl mpl i ollow. Lt r ow to b Gui itribut i ot ril i Gui. r Gui th joit it o i ot 0. Hr t th vlu o with qul probbilit. It i ot vr iiult to how tht or 30 tut tht i lo 0. Clrl mppig. Thror i δ o giv Gui itribut th Thror th itributio or tht Ppouli]. r rlt to h othr through rom r ot ipt. Furthrmor th oitio it δ giv. W how i R tht i r joitl i lw uivrit Gui itributio. giv whih i 0.5δ 0.5δ giv w r ot joitl Gui. For otrutiv mpl [Probbilit R.4 Momt Grtig Futio o joit Gui r.v. ro m [ ] Ipt C i qul to b ip. Th

8 Prpr b Çğt C MTU ltril gi. Dpt. 30 Sprig 0089 oumt vrio. Dpt C Lt Lt i th rom vribl b ipt ro m Gui rom vribl. rom ollow lt i rbitrr looig utio g ollow Hr g p p p p p r om lr. Wh th momt grtig utio or irt i th lt qutio w gt g i ow w itrprt g tblih it otio with. Th momt grtig utio i i or vlu or whih th pttio rult i iit. For uivrit Gui itributio uh utio i i or ll ompl vlu. Th m i lo vli or vribl. W th tht g utio i i or rbitrr ompl vlu pir o. Si th prmtr o g utio b rbitrr w hoo to rlbl thm to gt

9 Prpr b Çğt C MTU ltril gi. Dpt. 30 Sprig 0089 oumt vrio. p g w gt th ollowig or th momt grtig utio o. t w u th umtl thorm to pr th itributio o. W lo ivrt giv i to gt th itributio o but w prr uig th umtl thorm it o ivr Lpl trorm. Rwritig qutio i th mtri orm w gt th ollowig 0 0 Th p p A J 3 Hr J i th Jobi whih i J A i lr who t vlu i ot o itrt or ow. Fill w i giv i to om pil vlu. Th 3 bom p A 3

10 Prpr b Çğt C MTU ltril gi. Dpt. 30 Sprig 0089 oumt vrio. From 3 w gt th momt grtig utio o joitl itribut Gui ro m rom vribl. R.5 Momt Grtig Futio o joit Gui r.v. o-ro m Lt grtig utio o vribl b joitl itribut ro m Gui rom vribl. Th momt Th momt grtig utio o i giv i R.4 qutio. Di w rom b writt ollow whih i ot tht wh 0 th momt grtig utio giv i th lt qutio ru to th o ou or th ipt vribl i R.4. R.6 Corrltio oiit btw i. W ow tht th orrltio oiit btw rom vribl iitio i ot t b th m vlu o b it. W lo ow tht 0. W t th momt grtig utio o joit Gui rom vribl with ro m lult it prtil rivtiv ollow

11 Prpr b Çğt C MTU ltril gi. Dpt. 30 Sprig 0089 oumt vrio. Wh ro i ubtitut or w gt rom hr w tht. ot tht right h i o th lt rltio i th iitio o orrltio oiit or ro m rom vribl. Thror pprig i th iitio o th joit it i th tul vlu o orrltio oiit btw. W ll thi prmtr th orrltio oiit. R.7 I th orrltio oiit o joitl Gui itribut r.v. i ro th rom vribl r ipt. I R.5 w hv ot tht ot tht wh 0 th momt grtig utio giv i th lt qutio ru to th o ou or th ipt vribl i R.4. w hv lo how tht th orrltio oiit btw i R.5. Thi importt rult ollow rom th ombitio o R.4 R.5. R.7 Th mrgil it or i Gui. Th momt grtig utio or th mrgil it or i. Th m utio b pr rom th momt grtig utio o th joit itributio b tig 0. Wh thi i o w gt whih i th momt grtig utio o uivrit Gui itributio with m vri. H wh w mrgili joit Gui it th w hv uivrit Gui it. R.8 Th oitiol it o giv \ Thi i lo Gui itribut to b how

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