Gauss q-taqsimoti - Gaussian q-distribution
Ushbu maqola Diaz va Teruel tomonidan taqdim etilgan tarqatish haqida. Tsallis q-Gaussian uchun qarang
q-gauss .
Yilda matematik fizika va ehtimollik va statistika , Gauss q - tarqatish oila ehtimollik taqsimoti Bunga quyidagilar kiradi cheklovchi holatlar , bir xil taqsimlash va normal (Gauss) taqsimoti . Bu Diaz va Teruel tomonidan kiritilgan,[tushuntirish kerak a q-analog Gauss yoki normal taqsimot .
Tarqatish nolga teng nosimmetrik va normal taqsimotning chegaralangan holatidan tashqari chegaralangan. Cheklovli bir xil taqsimot -1 dan +1 gacha.
Ta'rif
Gauss q zichligi.
Ruxsat bering q bo'lishi a haqiqiy raqam [0, 1) oralig'ida. The ehtimollik zichligi funktsiyasi Gaussning q -taqsimlash tomonidan beriladi
s q ( x ) = { 0 agar x < − ν 1 v ( q ) E q 2 − q 2 x 2 [ 2 ] q agar − ν ≤ x ≤ ν 0 agar x > ν . { displaystyle s_ {q} (x) = { begin {case} 0 & { text {if}} x <- nu { frac {1} {c (q)}} E_ {q ^ { 2}} ^ { frac {-q ^ {2} x ^ {2}} {[2] _ {q}}} & { text {if}} - nu leq x leq nu 0 & { mbox {if}} x> nu. End {case}}} qayerda
ν = ν ( q ) = 1 1 − q , { displaystyle nu = nu (q) = { frac {1} { sqrt {1-q}}},} v ( q ) = 2 ( 1 − q ) 1 / 2 ∑ m = 0 ∞ ( − 1 ) m q m ( m + 1 ) ( 1 − q 2 m + 1 ) ( 1 − q 2 ) q 2 m . { displaystyle c (q) = 2 (1-q) ^ {1/2} sum _ {m = 0} ^ { infty} { frac {(-1) ^ {m} q ^ {m ( m + 1)}} {(1-q ^ {2m + 1}) (1-q ^ {2}) _ {q ^ {2}} ^ {m}}}.} The q - analog [t ]q t { displaystyle t}
[ t ] q = q t − 1 q − 1 . { displaystyle [t] _ {q} = { frac {q ^ {t} -1} {q-1}}.} The q - ning analogi eksponent funktsiya bo'ladi q-eksponent , E x q
E q x = ∑ j = 0 ∞ q j ( j − 1 ) / 2 x j [ j ] ! { displaystyle E_ {q} ^ {x} = sum _ {j = 0} ^ { infty} q ^ {j (j-1) / 2} { frac {x ^ {j}} {[j ]!}}} qaerda q - ning analogi faktorial bo'ladi q-faktorial , [n ]q
[ n ] q ! = [ n ] q [ n − 1 ] q ⋯ [ 2 ] q { displaystyle [n] _ {q}! = [n] _ {q} [n-1] _ {q} cdots [2] _ {q} ,} butun son uchun n > 2 va [1]q q
Kommulyativ Gauss q-taqsimoti.
The kümülatif taqsimlash funktsiyasi Gaussning q -taqsimlash tomonidan beriladi
G q ( x ) = { 0 agar x < − ν 1 v ( q ) ∫ − ν x E q 2 − q 2 t 2 / [ 2 ] d q t agar − ν ≤ x ≤ ν 1 agar x > ν { displaystyle G_ {q} (x) = { begin {case} 0 & { text {if}} x <- nu [12pt] displaystyle { frac {1} {c (q)}} int _ {- nu} ^ {x} E_ {q ^ {2}} ^ {- q ^ {2} t ^ {2} / [2]} , d_ {q} t & { text {if }} - nu leq x leq nu [12pt] 1 & { text {if}} x> nu end {case}}} qaerda integratsiya belgisi "." ni bildiradi Jekson ajralmas .
Funktsiya G q
G q ( x ) = { 0 agar x < − ν , 1 2 + 1 − q v ( q ) ∑ n = 0 ∞ q n ( n + 1 ) ( q − 1 ) n ( 1 − q 2 n + 1 ) ( 1 − q 2 ) q 2 n x 2 n + 1 agar − ν ≤ x ≤ ν 1 agar x > ν { displaystyle G_ {q} (x) = { begin {case} 0 & { text {if}} x <- nu, displaystyle { frac {1} {2}} + { frac { 1-q} {c (q)}} sum _ {n = 0} ^ { infty} { frac {q ^ {n (n + 1)} (q-1) ^ {n}} {( 1-q ^ {2n + 1}) (1-q ^ {2}) _ {q ^ {2}} ^ {n}}} x ^ {2n + 1} & { text {if}} - nu leq x leq nu 1 & { text {if}} x> nu end {case}}} qayerda
( a + b ) q n = ∏ men = 0 n − 1 ( a + q men b ) . { displaystyle (a + b) _ {q} ^ {n} = prod _ {i = 0} ^ {n-1} (a + q ^ {i} b).} Lahzalar
The lahzalar Gaussning q - tarqatish tomonidan beriladi
1 v ( q ) ∫ − ν ν E q 2 − q 2 x 2 / [ 2 ] x 2 n d q x = [ 2 n − 1 ] ! ! , { displaystyle { frac {1} {c (q)}} int _ {- nu} ^ { nu} E_ {q ^ {2}} ^ {- q ^ {2} x ^ {2} / [2]} , x ^ {2n} , d_ {q} x = [2n-1] !!,} 1 v ( q ) ∫ − ν ν E q 2 − q 2 x 2 / [ 2 ] x 2 n + 1 d q x = 0 , { displaystyle { frac {1} {c (q)}} int _ {- nu} ^ { nu} E_ {q ^ {2}} ^ {- q ^ {2} x ^ {2} / [2]} , x ^ {2n + 1} , d_ {q} x = 0,} bu erda belgi [2n - 1] !! bo'ladi q - ning analogi ikki faktorial tomonidan berilgan
[ 2 n − 1 ] [ 2 n − 3 ] ⋯ [ 1 ] = [ 2 n − 1 ] ! ! . { displaystyle [2n-1] [2n-3] cdots [1] = [2n-1] !!. ,} Shuningdek qarang
Adabiyotlar
Dias, R .; Pariguan, E. (2009). "Gauss q-taqsimoti to'g'risida". Matematik tahlil va ilovalar jurnali 358 : 1. arXiv :0807.1918 doi :10.1016 / j.jmaa.2009.04.046 . Diaz, R .; Teruel, C. (2005). "q, k-umumiy gamma va beta-funktsiyalar" (PDF) . Lineer bo'lmagan matematik fizika jurnali 12 (1): 118–134. arXiv :matematik / 0405402 Bibcode :2005 yil JNMP ... 12..118D . doi :10.2991 / jnmp.2005.12.1.10 . van Liven, X.; Maassen, H. (1995). "A q Gauss taqsimotining deformatsiyasi " (PDF) . Matematik fizika jurnali 36 (9): 4743. Bibcode :1995 yil JMP .... 36.4743V . CiteSeerX 10.1.1.24.6957 doi :10.1063/1.530917 . Exton, H. (1983), q-gipergeometrik funktsiyalar va ilovalar , Nyu-York: Halstead Press, Chichester: Ellis Xorvud, 1983, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538
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![{ displaystyle G_ {q} (x) = { begin {case} 0 & { text {if}} x <- nu [12pt] displaystyle { frac {1} {c (q)}} int _ {- nu} ^ {x} E_ {q ^ {2}} ^ {- q ^ {2} t ^ {2} / [2]} , d_ {q} t & { text {if }} - nu leq x leq nu [12pt] 1 & { text {if}} x> nu end {case}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2127f2fe582b05819f37838de70f0b976b4a75d6)
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