What Is the Difference between Gamma and Gaussian Distributions?

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1 Applid Mahmaics,,, hp://ddoiorg/6/am Publishd Oli Fbruary (hp://wwwscirporg/joural/am) Wha Is h Diffrc bw Gamma ad Gaussia Disribuios? iao-li Hu chool of Elcrical Egirig ad Compur cic, Uivrsiy of Nwcasl, Nwcasl, Ausralia iaolihu@wcsalduau, lhu@amssacc Rcivd Novmbr 9, ; rvisd Dcmbr 5, ; accpd Jauary, ABTRACT A iqualiy dscribig h diffrc bw Gamma ad Gaussia disribuios is drivd Th asympoic boud is much br ha by isig uiform boud from Brry-Ess iqualiy Kywords: Gamma Disribuio; Gaussia Disribuio; Brry-Ess Iqualiy; Characrisic Fucio Iroducio Problm W firs iroduc som oaios Do Gamma disribuio fucio as, d () for ad, whr is h Gamma fucio, i, d Assum, for Th dsiy of chisquar disribud radom variabl wih dgrs of frdom is,for, f,, ohrwis I is wll-ow ha h radom variabl ca b i- rprd by wih idpd ad id ically disribud (iid) radom variabls,,,,, whr, dos h sadard Gaussia disribuio Th ma ad variac of is rspcivly E, E Th, by simpl chag of variabl w fid P, () O h ohr sid, by h Brry-Ess iqualiy o,,,, i is asy o fid a boud C such ha whr io, i, C P, () is h sadard Gaussia disribuio fuc- d () π Th, by Equaios () ad () i follows, C, which dscribs h disac bw Gamma ad Gaussia disribuios Th purpos of his papr is o driv asympoic sharpr boud C i Equaio (5), which much improvs h cosa C by dircly usig Brry-Ess iqualiy Th mai framwor of aalysis is basd o Gil-Pla formula (ssially quival o Lvy ivrsio formula), which rprss disribuio fucio of a radom variabl by is characrisic fucio Th mai rsul of his papr is as followig Thorm A rlaio of h Gamma disribuio () ad Gaussia disribuio () is giv by whr C sup,, C C, π (5) (6) Copyrigh cirs

2 86 -L HU C π π 6 π π π 6 wih ad for ay, Clarly, C as Thus, h asympoical boud is C 88 π as To chc h ighss of h limi valu of C, w plo i Figur h muliplicaio sup, for,,,, whr h sraigh li is h limi valu π From his prim i sms ha π is h bs cosa Th dcy of h horical formula C is plod for, i Figur, which also shows h dcy o h limi valu Th slow rd is du o ha som uppr bouds π formulad ovr irval, hav b waly simad, g, h hird ad fourh rms of C Compariso o h Boud Drivd by Brry-Ess Iqualiy L,, b a squc of idpd idi- cally disribud radom variabls wih E = E Figur Eprim Figur Trd of C ad fii hird absolu mom E Do F P By classic Brry-Ess iqualiy, hr iss a fii posiiv umbr C such ha C df, sup F (7) Th bs uppr boud C 785 is foud i [] i 9 Th boud is improvd i [] a som agl i a sligh diffr form as, C d F (8) wih C mi77 9, Th iqualiy (8) will b sharpr ha Equaio (7) for 9 Now l us driv h cosa C i (5) by applyig Brry-Ess iqualiy o,,, I is difficul o calcula h ac valu of hird absolu mom of h radom variabl Thus, i is approimad as E 7 by usig Malab o igra ovr irval, dividd quivally, subirval for is half valu By Equaio (7) wih C 785 w hav C 75 ad by Equaio (8) w hav Copyrigh cirs

3 -L HU 87 C 7 Hc, h bs cosa C i Equaio (5) by applyig Brry-Ess iqualiy is 7 Obviously, h limi boud lim C 88 π foud i his papr for chi-squar disribuio is much br Th chical raso is ha h Brry-Ess iqualiy dals wih gral iid radom squcs wihou ac iformaio of h disribuio Proof of Mai Rsul Bfor o prov h mai rsul, w firs lis a fw lmmas ad iroduc som facs of characrisic fucio hory om Lmmas Lmma For a compl umbr Proof Firs show ha saisfyig, By Taylor s pasio ad oig, w hav Toghr wih!!, h assrio follows Lmma For a ral umbr saisfyig <, whr i Clarly, i ii p R is h imagiary ui ad R, i 6 8 R Proof By Taylor pasio for compl fucio, for < w hav i i i i i l i i i i i i R, whr R is show abov By furhr oig h wo alraig ral sris abov, i follows h uppr boud W ci blow a wll-ow iqualiy [] as a lmma Lmma Th ail probabiliy of h sadard ormal disribuio saisfis d π π π for Characrisic Fucio L us rcall, s g, [], h dfiiio ad som basic facs of characrisic fucio (CF), which provids aohr way o dscrib h disribuio fucio of a radom variabl Th characrisic fucio of a radom variabl is dfid by E i, whr i is h imagiary ui, ad R is h argum of h fucio Clarly, h CF for radom variabl Y a b wih ral umbrs a ad b is Aohr basic qualiy is Y a i Z Y b for Z Y wih ad Y idpd o ach ohr I is wll-ow ha h CF of sadard Gaussia, is (9) ad h CF of chi-squar disribud variabl is Thus, h CF for i is i i () Th CF is acually a ivrs Fourir rasformaio Copyrigh cirs

4 88 -L HU of dsiy fucio Thrfor, disribuio fucio ca b prssd by CF dircly, g, Lvy ivrsio formula W us aohr slighly simplr formula For a uivaria radom variabl, if is a coiuiy poi of is disribuio F, h i i F d, () πi which is calld Gil-Pla formula, s, g, pag 68 of [] Proof of Mai Rsul W ar ow i a posiio o prov h mai rsul Proof of Thorm Firs aaly CF of giv by Equaio () Do For <, i,, by Lmma, whr Clarly, To ma sur do R i i i p R, i R R i i R 6 R () for som,5 Th, i is asy o s ha, () for Hc, by Equaios () ad () ad Lmma, R R R R () for Now l us cosidr h diffrc bw ad, i, h CF (9) of Gaussia disribuio, ovr h irval, By Equaio () i πi d R d d π π d d π No ha i follows imilarly, π π d, d, i πi d 6 π π i πi d 6 π π (5) (6) Blow l us aaly h rsidual igrals ovr h irval By Lmma,, imilarly, i πi d d π d π π π (7) i d (8) πi π I is somwha difficul o aaly h rsidual igral Copyrigh cirs

5 -L HU 89 ovr, for W divid i io wo subirvals as followig: i d i d d πi π π whr d I I, π Obsrv ha dcrass o irval, ad for, w hav πi, whr 6 Th fac is usd i abov formula Thus, I, (9) 6 π For h ohr irval πi, d d, w procd as d By Equaios (9) ad () () imilarly, d () π i πi 6 π d () π i πi 6 π By Equaio (5), Equaio (7), Equaio () ad Equaio (6), Equaio (8), Equaio () whr i i i i d πi πi d C πi d, C C, π C π π π π π 6 I viw of Formula (), h formula o b provd follows dircly REFERENCE [] I Tyuri, O h Accuracy of h Gaussia Approimaio, Dolady Mahmaics, Vol 8, No, 9, pp 8-8 doi:/ [] V Korolva ad I hvsova, A Improvm of h Brry-Ess i Equaliy wih Applicaio o Possio ad Mid Poiso Radom ums, cadiavia Acuarial Joural, Vol, No,, pp 8-5 doi:8/68857 [] R D Gordo, Valus of Mills Raio of Ara o Boudig Ordia ad of h Normal Probabiliy Igral for Larg Valus of h Argum, Th Aals of Mahmaical aisics, Vol, No, 9, pp 6-66 doi:/aoms/7777 [] K L Chug, A Cours i Probabiliy Thory, rd Ediio, Probabiliy ad Mahmaical aisics, Acadmic, Nw Yor, Copyrigh cirs