# The q-gaussian Distribution Density, distribution function, quantile function and random generation for the q-gaussian distribution with parameters mu and sig. ```r dqgauss(x, q = 0, mu = 0, sig = 1) ``` ## Arguments - `x`: vector of quantiles. - `q`: entropic index. - `mu`: a value for q-mean. - `sig`: a value for q-variance. ## Returns dqgauss gives the density, pqgauss gives the distribution function, cqgauss gives the quantile function, and rqgauss generates random deviates. ## Details If q , mu and sig values are not specified, they assume the default values of 0, 0 and 1, respectively. Defining Z=(q-1)/(3-q), the q-gaussian distribution has density wrinten as p(x) = (sig*Beta(alpha/2,1/2))^-1*(1+Z(x-mu)^2/sig^2)^-(1+1/Z)/2 where alpha = 1 - 1/Z when q < 1 and 1/Z when 1 < q < 3. ## Examples ```r qv <- c(2.8,2.5,2,1.01,0,-5); nn <- 700 xrg <- sqrt((3-qv[6])/(1-qv[6])) xr <- seq(-xrg,xrg,by=2*xrg/nn) y0 <- dqgauss(xr,qv[6]) plot(xr,y0,ty='l',xlim=range(-4.5,4.5),ylab='p(x)',xlab='x') for (i in 1:5){ if (qv[i]< 1) xrg <- sqrt((3-qv[i])/(1-qv[i])) else xrg <- 4.5 vby <- 2*xrg/nn xr <- seq(-xrg,xrg,by=2*xrg/nn) y0 <- dqgauss(xr,qv[i]) points (xr,y0,ty='l',col=(i+1)) } legend(2, 0.4, legend =c(expression(paste(q==-5)),expression(paste(q==0)), expression(paste(q==1.01)),expression(paste(q==2)),expression(paste(q==2.5)), expression(paste(q==2.8))),col = c(1,6,5,4,3,2), lty = c(1,1,1,1,1,1)) ###### qv <- 0 rr <- rqgauss(2^16,qv) nn <- 70 xrg <- sqrt((3-qv)/(1-qv)) vby <- 2*xrg/(nn) xr <- seq(-xrg,xrg,by=vby) hist (rr,breaks=xr,freq=FALSE,xlab="x",main='') y <- dqgauss(xr) lines(xr,y/sum(y*vby),cex=.5,col=2,lty=4) ``` ## Author(s) Emerson Luis de Santa Helena , Wagner Santos de Lima ## References Thistleton, W.,Marsh, J. A.,Nelson, K.,Tsallis, C., (2007) IEEE Transactions on Information Theory, 53(12):4805 Tsallis, C., (2009) Introduction to Nonextensive Statistical Mechanics. Springer. de Santa Helena, E. L., Nascimento, C. M., and Gerhardt, G. J., (2015) Alternative way to charecterize a q-gaussian distribution by a robust heavy tail measurement. Physica A, (435):44-50. Manuscript submitted for publication (2016) qGaussian: Tools to Explore Applications of Tsallis Statistics ## See Also Distributions for other standard distributions, including dt and dcauchy. `Distributions`

dqgauss function