V teoriyi imovirnosti i statistici rozpodil Laplasa nalezhit do sim yi neperervnih rozpodiliv Nazvanij na chest francuzkogo matematika P yera Simona Laplasa Inkoli vzhivayut nazvu podvijnij eksponencijnij rozpodil mayuchi na uvazi sho grafik shilnosti rozpodilu Laplasa viglyadaye yak simetrichno prodovzhena na vid yemnij pivosi shilnist eksponencijnogo rozpodilu Rozpodil LaplasaFunkciya rozpodilu jmovirnostejParametri m displaystyle mu parametr lokalizaciyi dijsnij b gt 0 displaystyle b gt 0 parametr masshtabu dijsnij Nosij funkciyi x 0 displaystyle x in 0 infty Rozpodil imovirnostej 1 2 b exp x m b displaystyle frac 1 2 b exp left frac x mu b right Funkciya rozpodilu jmovirnostej cdf Divitsya tekstSerednye m displaystyle mu Mediana m displaystyle mu Moda m displaystyle mu Dispersiya 2 b 2 displaystyle 2 b 2 Koeficiyent asimetriyi 0 displaystyle 0 Koeficiyent ekscesu 3 displaystyle 3 Entropiya log 2 e b displaystyle log 2 e b Tvirna funkciya momentiv mgf exp m t 1 b 2 t 2 displaystyle frac exp mu t 1 b 2 t 2 dlya t lt 1 b displaystyle t lt 1 b Harakteristichna funkciya exp m i t 1 b 2 t 2 displaystyle frac exp mu i t 1 b 2 t 2 Riznicya znachen dvoh nezalezhnih odnakovo rozpodilenih eksponencijnih vipadkovih velichin rozpodilena za rozpodilom Laplasa takozh Brounivskij ruh v eksponencijno rozpodilenij tochci chasu rozpodilenij za Laplasom Zmist 1 Harakteristiki 1 1 Shilnist rozpodilu 1 2 Funkciya rozpodilu 2 Matematichne spodivannya i dispersiya 3 Momenti 4 Generaciya vipadkovih velichin rozpodilenih za LaplasomHarakteristiki RedaguvatiShilnist rozpodilu Redaguvati Vipadkova velichina rozpodilena z rozpodilom Laplasa X Lap m b maye shilnist f x m b 1 2 b exp x m b 1 2 b exp m x b x lt m exp x m b x m displaystyle f x mu b frac 1 2b exp left frac x mu b right frac 1 2b left begin matrix exp left frac mu x b right amp mbox x lt mu 8pt exp left frac x mu b right amp mbox x geq mu end matrix right nbsp de m koeficiyent zsuvu i b gt 0 koeficiyent masshtabu Yaksho m 0 i b 1 to dodatna pivvis ce eksponencijnij rozpodil pomnozhenij na 1 2 displaystyle frac 1 2 nbsp Shilnist rozpodilu Laplasa nagaduye shilnist normalnogo rozpodilu z tiyeyu vidminnistyu sho viraz shilnosti normalnogo rozpodilu mistit kvadrat riznici znachennya i matematichnogo spodivannya m a u virazi dlya shilnosti Laplasovogo rozpodilu modul ciyeyi riznici Yak naslidok Laplasiv rozpodil maye tovstishi hvosti nizh normalnij rozpodil Funkciya rozpodilu Redaguvati Funkciyu rozpodilu legko otrimati prointegruvavshi shilnist i vikoristovuyuchi simetrichnist shilnosti vidnosno parametra m Funkciya rozpodilu maye viglyad F x x f u d u 1 2 exp x m b x lt m 1 1 2 exp x m b x m displaystyle F x int infty x f u mathrm d u left begin matrix amp frac 1 2 exp left frac x mu b right amp mbox x lt mu 8pt 1 amp frac 1 2 exp left frac x mu b right amp mbox x geq mu end matrix right nbsp 0 5 1 sgn x m 1 exp x m b displaystyle 0 5 1 operatorname sgn x mu 1 exp x mu b nbsp Obernene do funkciyi rozpolu zapisuyetsya F 1 p m b sgn p 0 5 ln 1 2 p 0 5 displaystyle F 1 p mu b operatorname sgn p 0 5 ln 1 2 p 0 5 nbsp Matematichne spodivannya i dispersiya RedaguvatiV pokazniku eksponenti shilnosti mayemo modul riznici tomu interval displaystyle infty infty nbsp neobhidno rozbiti na b displaystyle infty beta nbsp i b displaystyle beta infty nbsp funkciya shilnosti simetrichna vidnosno cih intervaliv Integrali berutsya chastinami pri pidstanovci neskinchenostej displaystyle pm infty nbsp rozglyadayemo granici viglyadu lim x r x displaystyle lim x to pm infty r x nbsp E 3 x f x d x a 2 b x e a x b d x a 2 b x e a x b d x displaystyle operatorname E xi int limits infty infty xf x dx frac alpha 2 int limits infty beta xe alpha x beta dx frac alpha 2 int limits beta infty xe alpha x beta dx nbsp a 2 1 a x e a x b b a 2 1 a b e a x b d x a 2 1 a x e a x b b a 2 1 a b e a x b d x displaystyle frac alpha 2 frac 1 alpha xe alpha x beta bigg infty beta frac alpha 2 frac 1 alpha int limits infty beta e alpha x beta dx frac alpha 2 frac 1 alpha xe alpha x beta bigg beta infty frac alpha 2 frac 1 alpha int limits beta infty e alpha x beta dx nbsp b 2 1 2 a e a x b b b 2 1 2 a e a x b b b 1 2 a 1 2 a b displaystyle frac beta 2 frac 1 2 alpha e alpha x beta bigg infty beta frac beta 2 frac 1 2 alpha e alpha x beta bigg beta infty beta frac 1 2 alpha frac 1 2 alpha beta nbsp E 3 2 x 2 f x d x a 2 b x 2 e a x b d x a 2 b x 2 e a x b d x displaystyle operatorname E xi 2 int limits infty infty x 2 f x dx frac alpha 2 int limits infty beta x 2 e alpha x beta dx frac alpha 2 int limits beta infty x 2 e alpha x beta dx nbsp a 2 x 2 e a x b a b a 2 2 a b x e a x b d x a 2 x 2 e a x b a b a 2 2 a b x e a x b d x b 2 2 b a 1 a 2 b 2 2 b a 1 a 2 b 2 2 a 2 displaystyle frac alpha 2 frac x 2 e alpha x beta alpha bigg infty beta frac alpha 2 frac 2 alpha int limits infty beta xe alpha x beta dx frac alpha 2 frac x 2 e alpha x beta alpha bigg beta infty frac alpha 2 frac 2 alpha int limits beta infty xe alpha x beta dx frac beta 2 2 frac beta alpha frac 1 alpha 2 frac beta 2 2 frac beta alpha frac 1 alpha 2 beta 2 frac 2 alpha 2 nbsp D 3 E 3 2 E 3 2 b 2 2 a 2 b 2 2 a 2 displaystyle operatorname D xi operatorname E xi 2 operatorname E xi 2 beta 2 frac 2 alpha 2 beta 2 frac 2 alpha 2 nbsp Momenti RedaguvatiE 3 k x k f x d x a 2 b x k e a x b d x a 2 b x k e a x b d x displaystyle operatorname E xi k int limits infty infty x k f x dx frac alpha 2 int limits infty beta x k e alpha x beta dx frac alpha 2 int limits beta infty x k e alpha x beta dx nbsp Zastosovuyuchi formulu integruvannya chastinami dekilka raz otrimuyemo x k e a x b d x 1 a x k e a x b k a 2 x k 1 e a x b k k 1 a 3 x k 2 e a x b displaystyle int x k e alpha x beta dx frac 1 alpha x k e alpha x beta frac k alpha 2 x k 1 e alpha x beta frac k k 1 alpha 3 x k 2 e alpha x beta nbsp 1 k 1 k k 1 3 2 a k x e a x b 1 k k k 1 2 1 a k 1 e a x b displaystyle ldots 1 k 1 frac k k 1 cdots 3 cdot 2 alpha k xe alpha x beta 1 k frac k k 1 cdots 2 cdot 1 alpha k 1 e alpha x beta nbsp x k e a x b d x 1 a x k e a x b k a 2 x k 1 e a x b k k 1 a 3 x k 2 e a x b displaystyle int x k e alpha x beta dx frac 1 alpha x k e alpha x beta frac k alpha 2 x k 1 e alpha x beta frac k k 1 alpha 3 x k 2 e alpha x beta nbsp k k 1 3 2 a k x e a x b k k 1 2 1 a k 1 e a x b displaystyle ldots frac k k 1 cdots 3 cdot 2 alpha k xe alpha x beta frac k k 1 cdots 2 cdot 1 alpha k 1 e alpha x beta nbsp Pislya pidstanovok granic integruvannya b x k e a x b d x 1 a b k k a 2 b k 1 k k 1 a 3 b k 2 displaystyle int limits infty beta x k e alpha x beta dx frac 1 alpha beta k frac k alpha 2 beta k 1 frac k k 1 alpha 3 beta k 2 nbsp 1 k 1 k k 1 3 2 a k b 1 k k k 1 2 1 a k 1 displaystyle ldots 1 k 1 frac k k 1 cdots 3 cdot 2 alpha k beta 1 k frac k k 1 cdots 2 cdot 1 alpha k 1 nbsp b x k e a x b d x 1 a b k k a 2 b k 1 k k 1 a 3 b k 2 k k 1 3 2 a k b k k 1 2 1 a k 1 displaystyle int limits beta infty x k e alpha x beta dx frac 1 alpha beta k frac k alpha 2 beta k 1 frac k k 1 alpha 3 beta k 2 ldots frac k k 1 cdots 3 cdot 2 alpha k beta frac k k 1 cdots 2 cdot 1 alpha k 1 nbsp Oskilki pershij integral zalezhit vid parnosti k rozglyadayutsya dvavipadki k parne i k neparne E 3 k b k k k 1 a 2 b k 2 k k 1 2 1 a k k 2 n b k k k 1 a 2 b k 2 k k 1 3 2 a k 1 b k 2 n 1 displaystyle operatorname E xi k begin cases beta k frac k k 1 alpha 2 beta k 2 ldots frac k k 1 cdots 2 cdot 1 alpha k amp k 2n beta k frac k k 1 alpha 2 beta k 2 ldots frac k k 1 cdots 3 cdot 2 alpha k 1 beta amp k 2n 1 end cases nbsp Abo v zagalnomu viglyadi E 3 k i 0 k 2 b k 2 i a 2 i k k 2 i displaystyle operatorname E xi k sum i 0 left lfloor k 2 right rfloor frac beta k 2i alpha 2i frac k k 2i nbsp de x displaystyle left lfloor x right rfloor nbsp cila chastina x Generaciya vipadkovih velichin rozpodilenih za Laplasom RedaguvatiNehaj mayemo vipadkovu velichinu U rivnomirno rozpodilenu na intervali 1 2 1 2 todi vipadkova velichina X m b sgn U ln 1 2 U displaystyle X mu b operatorname sgn U ln 1 2 U nbsp rozpodilena za rozpodilom Laplasa z parametrami m and b Ce vidno yaksho rozglyanuti funkciyu obernenu do funkciyi rozpodilu yaka navedena vishe Vipadkovu velichinu X displaystyle X nbsp L a p 0 b displaystyle Lap 0 b nbsp mozhna takozh zgeneruvati yak riznicyu dvoh n o r E x p 1 b displaystyle Exp 1 b nbsp vipadkovih velichin Abo she vipadkovu velichinu X displaystyle X nbsp L a p 0 1 displaystyle Lap 0 1 nbsp mozhna zgeneruvati yak logarifm chastki dvoh n o r rivnomirno rozpodilenih vipadkovih velichin Otrimano z https uk wikipedia org w index php title Rozpodil Laplasa amp oldid 36155904
