Family of probability distributions
This article is about the distribution introduced by Diaz and Teruel. For the Tsallis q-Gaussian, see
q-Gaussian .
Inmathematical physics andprobability andstatistics , theGaussianq -distribution is a family ofprobability distributions that includes, aslimiting cases , theuniform distribution and thenormal (Gaussian) distribution . It was introduced by Diaz and Teruel.[clarification needed It is aq-analog of the Gaussian ornormal distribution .
The distribution is symmetric about zero and is bounded, except for the limiting case of the normal distribution. The limiting uniform distribution is on the range -1 to +1.
The Gaussian q-density. Letq be areal number in the interval [0, 1). Theprobability density function of the Gaussianq -distribution is given by
s q ( x ) = { 0 if x < − ν 1 c ( q ) E q 2 − q 2 x 2 [ 2 ] q if − ν ≤ x ≤ ν 0 if x > ν . {\displaystyle s_{q}(x)={\begin{cases}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{cases}}} where
ν = ν ( q ) = 1 1 − q , {\displaystyle \nu =\nu (q)={\frac {1}{\sqrt {1-q}}},} c ( 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}}}.} Theq -analogue [t ]q t {\displaystyle t}
[ t ] q = q t − 1 q − 1 . {\displaystyle [t]_{q}={\frac {q^{t}-1}{q-1}}.} Theq -analogue of theexponential function is theq-exponential ,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]!}}} where theq -analogue of thefactorial is theq-factorial , [n ]q
[ n ] q ! = [ n ] q [ n − 1 ] q ⋯ [ 2 ] q {\displaystyle [n]_{q}!=[n]_{q}[n-1]_{q}\cdots [2]_{q}\,} for an integern > 2 and [1]q q
The Cumulative Gaussian q-distribution. Thecumulative distribution function of the Gaussianq -distribution is given by
G q ( x ) = { 0 if x < − ν 1 c ( q ) ∫ − ν x E q 2 − q 2 t 2 / [ 2 ] d q t if − ν ≤ x ≤ ν 1 if x > ν {\displaystyle G_{q}(x)={\begin{cases}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{cases}}} where theintegration symbol denotes theJackson integral .
The functionG q
G q ( x ) = { 0 if x < − ν , 1 2 + 1 − q c ( 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 if − ν ≤ x ≤ ν 1 if x > ν {\displaystyle G_{q}(x)={\begin{cases}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{cases}}} where
( a + b ) q n = ∏ i = 0 n − 1 ( a + q i b ) . {\displaystyle (a+b)_{q}^{n}=\prod _{i=0}^{n-1}(a+q^{i}b).} Themoments of the Gaussianq -distribution are given by
1 c ( 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 c ( 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,} where the symbol [2n − 1]!! is theq -analogue of thedouble factorial given by
[ 2 n − 1 ] [ 2 n − 3 ] ⋯ [ 1 ] = [ 2 n − 1 ] ! ! . {\displaystyle [2n-1][2n-3]\cdots [1]=[2n-1]!!.\,} Díaz, R.; Pariguan, E. (2009). "On the Gaussian q-distribution".Journal of Mathematical Analysis and Applications 358 :1– 9.arXiv :0807.1918 doi :10.1016/j.jmaa.2009.04.046 .S2CID 115175228 . Diaz, R.; Teruel, C. (2005)."q,k-Generalized Gamma and Beta Functions" (PDF) .Journal of Nonlinear Mathematical Physics 12 (1):118– 134.arXiv :math/0405402 Bibcode :2005JNMP...12..118D .doi :10.2991/jnmp.2005.12.1.10 .S2CID 73643153 . van Leeuwen, H.; Maassen, H. (1995)."Aq deformation of the Gauss distribution" (PDF) .Journal of Mathematical Physics 36 (9): 4743.Bibcode :1995JMP....36.4743V .CiteSeerX 10.1.1.24.6957 doi :10.1063/1.530917 .hdl :2066/141604 .S2CID 13934946 . Exton, H. (1983),q-Hypergeometric Functions and Applications , New York: Halstead Press, Chichester: Ellis Horwood, 1983,ISBN 0853124914 ISBN 0470274530 ISBN 978-0470274538
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
