# set terminal canvas rounded size 600,400 enhanced fsize 10 lw 1.6 fontscale 1 name "prob2_7" jsdir "." # set output 'prob2.7.js' set format x "%2.0f" set format y "%3.2f" set key fixed right top vertical Right noreverse enhanced autotitle box lt black linewidth 1.000 dashtype solid set label 1 "mu" at 5.83333, 0.0170962, 0.00000 left norotate back nopoint set label 2 "sigma" at 8.81476, 0.0578888, 0.00000 left norotate back nopoint set arrow 1 from 5.33333, 0.00000, 0.00000 to 5.33333, 0.130380, 0.00000 nohead back nofilled linewidth 1.000 dashtype solid set arrow 2 from 5.33333, 0.0578888, 0.00000 to 8.31476, 0.0578888, 0.00000 nohead back nofilled linewidth 1.000 dashtype solid set samples 200, 200 set style data lines set ytics norangelimit 0.00000,0.0170962,0.170962 set title "negative binomial PDF using gamma approximation" set xlabel "k, x ->" set xrange [ -2.00000 : 19.0000 ] noreverse nowriteback set ylabel "probability density ->" set yrange [ 0.00000 : 0.170962 ] noreverse nowriteback isint(x)=(int(x)==x) Binv(p,q)=exp(lgamma(p+q)-lgamma(p)-lgamma(q)) arcsin(x,r)=r<=0?1/0:abs(x)>r?0.0:invpi/sqrt(r*r-x*x) beta(x,p,q)=p<=0||q<=0?1/0:x<0||x>1?0.0:Binv(p,q)*x**(p-1.0)*(1.0-x)**(q-1.0) binom(x,n,p)=p<0.0||p>1.0||n<0||!isint(n)?1/0: !isint(x)?1/0:x<0||x>n?0.0:exp(lgamma(n+1)-lgamma(n-x+1)-lgamma(x+1) +x*log(p)+(n-x)*log(1.0-p)) cauchy(x,a,b)=b<=0?1/0:b/(pi*(b*b+(x-a)**2)) chisq(x,k)=k<=0||!isint(k)?1/0: x<=0?0.0:exp((0.5*k-1.0)*log(x)-0.5*x-lgamma(0.5*k)-k*0.5*log2) erlang(x,n,lambda)=n<=0||!isint(n)||lambda<=0?1/0: x<0?0.0:x==0?(n==1?real(lambda):0.0):exp(n*log(lambda)+(n-1.0)*log(x)-lgamma(n)-lambda*x) extreme(x,mu,alpha)=alpha<=0?1/0:alpha*(exp(-alpha*(x-mu)-exp(-alpha*(x-mu)))) f(x,d1,d2)=d1<=0||!isint(d1)||d2<=0||!isint(d2)?1/0: Binv(0.5*d1,0.5*d2)*(real(d1)/d2)**(0.5*d1)*x**(0.5*d1-1.0)/(1.0+(real(d1)/d2)*x)**(0.5*(d1+d2)) gmm(x,rho,lambda)=rho<=0||lambda<=0?1/0: x<0?0.0:x==0?(rho>1?0.0:rho==1?real(lambda):1/0): exp(rho*log(lambda)+(rho-1.0)*log(x)-lgamma(rho)-lambda*x) geometric(x,p)=p<=0||p>1?1/0: !isint(x)?1/0:x<0||p==1?(x==0?1.0:0.0):exp(log(p)+x*log(1.0-p)) halfnormal(x,sigma)=sigma<=0?1/0:x<0?0.0:sqrt2invpi/sigma*exp(-0.5*(x/sigma)**2) hypgeo(x,N,C,d)=N<0||!isint(N)||C<0||C>N||!isint(C)||d<0||d>N||!isint(d)?1/0: !isint(x)?1/0:x>d||x>C||x<0||x<d-(N-C)?0.0:exp(lgamma(C+1)-lgamma(C-x+1)-lgamma(x+1)+ lgamma(N-C+1)-lgamma(d-x+1)-lgamma(N-C-d+x+1)+lgamma(N-d+1)+lgamma(d+1)-lgamma(N+1)) laplace(x,mu,b)=b<=0?1/0:0.5/b*exp(-abs(x-mu)/b) logistic(x,a,lambda)=lambda<=0?1/0:lambda*exp(-lambda*(x-a))/(1.0+exp(-lambda*(x-a)))**2 lognormal(x,mu,sigma)=sigma<=0?1/0: x<0?0.0:invsqrt2pi/sigma/x*exp(-0.5*((log(x)-mu)/sigma)**2) maxwell(x,a)=a<=0?1/0:x<0?0.0:fourinvsqrtpi*a**3*x*x*exp(-a*a*x*x) negbin(x,r,p)=r<=0||!isint(r)||p<=0||p>1?1/0: !isint(x)?1/0:x<0?0.0:p==1?(x==0?1.0:0.0):exp(lgamma(r+x)-lgamma(r)-lgamma(x+1)+ r*log(p)+x*log(1.0-p)) nexp(x,lambda)=lambda<=0?1/0:x<0?0.0:lambda*exp(-lambda*x) normal(x,mu,sigma)=sigma<=0?1/0:invsqrt2pi/sigma*exp(-0.5*((x-mu)/sigma)**2) pareto(x,a,b)=a<=0||b<0||!isint(b)?1/0:x<a?0:real(b)/x*(real(a)/x)**b poisson(x,mu)=mu<=0?1/0:!isint(x)?1/0:x<0?0.0:exp(x*log(mu)-lgamma(x+1)-mu) rayleigh(x,lambda)=lambda<=0?1/0:x<0?0.0:lambda*2.0*x*exp(-lambda*x*x) sine(x,f,a)=a<=0?1/0: x<0||x>=a?0.0:f==0?1.0/a:2.0/a*sin(f*pi*x/a)**2/(1-sin(twopi*f)) t(x,nu)=nu<0||!isint(nu)?1/0: Binv(0.5*nu,0.5)/sqrt(nu)*(1+real(x*x)/nu)**(-0.5*(nu+1.0)) triangular(x,m,g)=g<=0?1/0:x<m-g||x>=m+g?0.0:1.0/g-abs(x-m)/real(g*g) uniform(x,a,b)=x<(a<b?a:b)||x>=(a>b?a:b)?0.0:1.0/abs(b-a) weibull(x,a,lambda)=a<=0||lambda<=0?1/0: x<0?0.0:x==0?(a>1?0.0:a==1?real(lambda):1/0): exp(log(a)+a*log(lambda)+(a-1)*log(x)-(lambda*x)**a) carcsin(x,r)=r<=0?1/0:x<-r?0.0:x>r?1.0:0.5+invpi*asin(x/r) cbeta(x,p,q)=ibeta(p,q,x) cbinom(x,n,p)=p<0.0||p>1.0||n<0||!isint(n)?1/0: !isint(x)?1/0:x<0?0.0:x>=n?1.0:ibeta(n-x,x+1.0,1.0-p) ccauchy(x,a,b)=b<=0?1/0:0.5+invpi*atan((x-a)/b) cchisq(x,k)=k<=0||!isint(k)?1/0:x<0?0.0:igamma(0.5*k,0.5*x) cerlang(x,n,lambda)=n<=0||!isint(n)||lambda<=0?1/0:x<0?0.0:igamma(n,lambda*x) cextreme(x,mu,alpha)=alpha<=0?1/0:exp(-exp(-alpha*(x-mu))) cf(x,d1,d2)=d1<=0||!isint(d1)||d2<=0||!isint(d2)?1/0:1.0-ibeta(0.5*d2,0.5*d1,d2/(d2+d1*x)) cgmm(x,rho,lambda)=rho<=0||lambda<=0?1/0:x<0?0.0:igamma(rho,x*lambda) cgeometric(x,p)=p<=0||p>1?1/0: !isint(x)?1/0:x<0||p==0?0.0:p==1?1.0:1.0-exp((x+1)*log(1.0-p)) chalfnormal(x,sigma)=sigma<=0?1/0:x<0?0.0:erf(x/sigma/sqrt2) chypgeo(x,N,C,d)=N<0||!isint(N)||C<0||C>N||!isint(C)||d<0||d>N||!isint(d)?1/0: !isint(x)?1/0:x<0||x<d-(N-C)?0.0:x>d||x>C?1.0:hypgeo(x,N,C,d)+chypgeo(x-1,N,C,d) claplace(x,mu,b)=b<=0?1/0:(x<mu)?0.5*exp((x-mu)/b):1.0-0.5*exp(-(x-mu)/b) clogistic(x,a,lambda)=lambda<=0?1/0:1.0/(1+exp(-lambda*(x-a))) clognormal(x,mu,sigma)=sigma<=0?1/0:x<=0?0.0:cnormal(log(x),mu,sigma) cnormal(x,mu,sigma)=sigma<=0?1/0:0.5+0.5*erf((x-mu)/sigma/sqrt2) cmaxwell(x,a)=a<=0?1/0:x<0?0.0:igamma(1.5,a*a*x*x) cnegbin(x,r,p)=r<=0||!isint(r)||p<=0||p>1?1/0: !isint(x)?1/0:x<0?0.0:ibeta(r,x+1,p) cnexp(x,lambda)=lambda<=0?1/0:x<0?0.0:1-exp(-lambda*x) cpareto(x,a,b)=a<=0||b<0||!isint(b)?1/0:x<a?0.0:1.0-(real(a)/x)**b cpoisson(x,mu)=mu<=0?1/0:!isint(x)?1/0:x<0?0.0:1-igamma(x+1.0,mu) crayleigh(x,lambda)=lambda<=0?1/0:x<0?0.0:1.0-exp(-lambda*x*x) csine(x,f,a)=a<=0?1/0: x<0?0.0:x>a?1.0:f==0?real(x)/a:(real(x)/a-sin(f*twopi*x/a)/(f*twopi))/(1.0-sin(twopi*f)/(twopi*f)) ct(x,nu)=nu<0||!isint(nu)?1/0:0.5+0.5*sgn(x)*(1-ibeta(0.5*nu,0.5,nu/(nu+x*x))) ctriangular(x,m,g)=g<=0?1/0: x<m-g?0.0:x>=m+g?1.0:0.5+real(x-m)/g-(x-m)*abs(x-m)/(2.0*g*g) cuniform(x,a,b)=x<(a<b?a:b)?0.0:x>=(a>b?a:b)?1.0:real(x-a)/(b-a) cweibull(x,a,lambda)=a<=0||lambda<=0?1/0:x<0?0.0:1.0-exp(-(lambda*x)**a) rnd(x) = floor(x+0.5) GPFUN_isint = "isint(x)=(int(x)==x)" fourinvsqrtpi = 2.25675833419103 invpi = 0.318309886183791 invsqrt2pi = 0.398942280401433 log2 = 0.693147180559945 sqrt2 = 1.4142135623731 sqrt2invpi = 0.797884560802865 twopi = 6.28318530717959 GPFUN_Binv = "Binv(p,q)=exp(lgamma(p+q)-lgamma(p)-lgamma(q))" GPFUN_arcsin = "arcsin(x,r)=r<=0?1/0:abs(x)>r?0.0:invpi/sqrt(r*r-x*x)" GPFUN_beta = "beta(x,p,q)=p<=0||q<=0?1/0:x<0||x>1?0.0:Binv(p,q)*x**(p-1.0)*(1.0-x)**(q-1.0)" GPFUN_binom = "binom(x,n,p)=p<0.0||p>1.0||n<0||!isint(n)?1/0: !isint(x)?1/0:x<0||x>n?0.0:exp(lgamma(n+1)-lgamma(n-x+1)-lgamma(x+1) +x*log(p)+(n-x)*log(1.0-p))" GPFUN_cauchy = "cauchy(x,a,b)=b<=0?1/0:b/(pi*(b*b+(x-a)**2))" GPFUN_chisq = "chisq(x,k)=k<=0||!isint(k)?1/0: x<=0?0.0:exp((0.5*k-1.0)*log(x)-0.5*x-lgamma(0.5*k)-k*0.5*log2)" GPFUN_erlang = "erlang(x,n,lambda)=n<=0||!isint(n)||lambda<=0?1/0: x<0?0.0:x==0?(n==1?real(lambda):0.0):exp(n*log(lambda)+(n-1.0)*log(x)-lgamma(n)-lambda*x)" GPFUN_extreme = "extreme(x,mu,alpha)=alpha<=0?1/0:alpha*(exp(-alpha*(x-mu)-exp(-alpha*(x-mu))))" GPFUN_f = "f(x,d1,d2)=d1<=0||!isint(d1)||d2<=0||!isint(d2)?1/0: Binv(0.5*d1,0.5*d2)*(real(d1)/d2)**(0.5*d1)*x**(0.5*d1-1.0)/(1.0+(real(d1)/d2)*x)**(0.5*(d1+d2))" GPFUN_gmm = "gmm(x,rho,lambda)=rho<=0||lambda<=0?1/0: x<0?0.0:x==0?(rho>1?0.0:rho==1?real(lambda):1/0): exp(rho*log(lambda)+(rho-1.0)*log(x)-lgamma(rho)-lambda*x)" GPFUN_geometric = "geometric(x,p)=p<=0||p>1?1/0: !isint(x)?1/0:x<0||p==1?(x==0?1.0:0.0):exp(log(p)+x*log(1.0-p))" GPFUN_halfnormal = "halfnormal(x,sigma)=sigma<=0?1/0:x<0?0.0:sqrt2invpi/sigma*exp(-0.5*(x/sigma)**2)" GPFUN_hypgeo = "hypgeo(x,N,C,d)=N<0||!isint(N)||C<0||C>N||!isint(C)||d<0||d>N||!isint(d)?1/0: !isint(x)?1/0:x>d||x>C||x<0||x<d-(N-C)?0.0:exp(lgamma(C+1)-lgamma(C-x+1)-lgamma(x+1)+ lgamma(N-C+1)-lgamma(d-x+1)-lgamma(N-C-d+x+1)+lgamma(N-d+1)+lgamma(d+1)-lgamma(N+1))" GPFUN_laplace = "laplace(x,mu,b)=b<=0?1/0:0.5/b*exp(-abs(x-mu)/b)" GPFUN_logistic = "logistic(x,a,lambda)=lambda<=0?1/0:lambda*exp(-lambda*(x-a))/(1.0+exp(-lambda*(x-a)))**2" GPFUN_lognormal = "lognormal(x,mu,sigma)=sigma<=0?1/0: x<0?0.0:invsqrt2pi/sigma/x*exp(-0.5*((log(x)-mu)/sigma)**2)" GPFUN_maxwell = "maxwell(x,a)=a<=0?1/0:x<0?0.0:fourinvsqrtpi*a**3*x*x*exp(-a*a*x*x)" GPFUN_negbin = "negbin(x,r,p)=r<=0||!isint(r)||p<=0||p>1?1/0: !isint(x)?1/0:x<0?0.0:p==1?(x==0?1.0:0.0):exp(lgamma(r+x)-lgamma(r)-lgamma(x+1)+ r*log(p)+x*log(1.0-p))" GPFUN_nexp = "nexp(x,lambda)=lambda<=0?1/0:x<0?0.0:lambda*exp(-lambda*x)" GPFUN_normal = "normal(x,mu,sigma)=sigma<=0?1/0:invsqrt2pi/sigma*exp(-0.5*((x-mu)/sigma)**2)" GPFUN_pareto = "pareto(x,a,b)=a<=0||b<0||!isint(b)?1/0:x<a?0:real(b)/x*(real(a)/x)**b" GPFUN_poisson = "poisson(x,mu)=mu<=0?1/0:!isint(x)?1/0:x<0?0.0:exp(x*log(mu)-lgamma(x+1)-mu)" GPFUN_rayleigh = "rayleigh(x,lambda)=lambda<=0?1/0:x<0?0.0:lambda*2.0*x*exp(-lambda*x*x)" GPFUN_sine = "sine(x,f,a)=a<=0?1/0: x<0||x>=a?0.0:f==0?1.0/a:2.0/a*sin(f*pi*x/a)**2/(1-sin(twopi*f))" GPFUN_t = "t(x,nu)=nu<0||!isint(nu)?1/0: Binv(0.5*nu,0.5)/sqrt(nu)*(1+real(x*x)/nu)**(-0.5*(nu+1.0))" GPFUN_triangular = "triangular(x,m,g)=g<=0?1/0:x<m-g||x>=m+g?0.0:1.0/g-abs(x-m)/real(g*g)" GPFUN_uniform = "uniform(x,a,b)=x<(a<b?a:b)||x>=(a>b?a:b)?0.0:1.0/abs(b-a)" GPFUN_weibull = "weibull(x,a,lambda)=a<=0||lambda<=0?1/0: x<0?0.0:x==0?(a>1?0.0:a==1?real(lambda):1/0): exp(log(a)+a*log(lambda)+(a-1)*log(x)-(lambda*x)**a)" GPFUN_carcsin = "carcsin(x,r)=r<=0?1/0:x<-r?0.0:x>r?1.0:0.5+invpi*asin(x/r)" GPFUN_cbeta = "cbeta(x,p,q)=ibeta(p,q,x)" GPFUN_cbinom = "cbinom(x,n,p)=p<0.0||p>1.0||n<0||!isint(n)?1/0: !isint(x)?1/0:x<0?0.0:x>=n?1.0:ibeta(n-x,x+1.0,1.0-p)" GPFUN_ccauchy = "ccauchy(x,a,b)=b<=0?1/0:0.5+invpi*atan((x-a)/b)" GPFUN_cchisq = "cchisq(x,k)=k<=0||!isint(k)?1/0:x<0?0.0:igamma(0.5*k,0.5*x)" GPFUN_cerlang = "cerlang(x,n,lambda)=n<=0||!isint(n)||lambda<=0?1/0:x<0?0.0:igamma(n,lambda*x)" GPFUN_cextreme = "cextreme(x,mu,alpha)=alpha<=0?1/0:exp(-exp(-alpha*(x-mu)))" GPFUN_cf = "cf(x,d1,d2)=d1<=0||!isint(d1)||d2<=0||!isint(d2)?1/0:1.0-ibeta(0.5*d2,0.5*d1,d2/(d2+d1*x))" GPFUN_cgmm = "cgmm(x,rho,lambda)=rho<=0||lambda<=0?1/0:x<0?0.0:igamma(rho,x*lambda)" GPFUN_cgeometric = "cgeometric(x,p)=p<=0||p>1?1/0: !isint(x)?1/0:x<0||p==0?0.0:p==1?1.0:1.0-exp((x+1)*log(1.0-p))" GPFUN_chalfnormal = "chalfnormal(x,sigma)=sigma<=0?1/0:x<0?0.0:erf(x/sigma/sqrt2)" GPFUN_chypgeo = "chypgeo(x,N,C,d)=N<0||!isint(N)||C<0||C>N||!isint(C)||d<0||d>N||!isint(d)?1/0: !isint(x)?1/0:x<0||x<d-(N-C)?0.0:x>d||x>C?1.0:hypgeo(x,N,C,d)+chypgeo(x-1,N,C,d)" GPFUN_claplace = "claplace(x,mu,b)=b<=0?1/0:(x<mu)?0.5*exp((x-mu)/b):1.0-0.5*exp(-(x-mu)/b)" GPFUN_clogistic = "clogistic(x,a,lambda)=lambda<=0?1/0:1.0/(1+exp(-lambda*(x-a)))" GPFUN_clognormal = "clognormal(x,mu,sigma)=sigma<=0?1/0:x<=0?0.0:cnormal(log(x),mu,sigma)" GPFUN_cmaxwell = "cmaxwell(x,a)=a<=0?1/0:x<0?0.0:igamma(1.5,a*a*x*x)" GPFUN_cnegbin = "cnegbin(x,r,p)=r<=0||!isint(r)||p<=0||p>1?1/0: !isint(x)?1/0:x<0?0.0:ibeta(r,x+1,p)" GPFUN_cnexp = "cnexp(x,lambda)=lambda<=0?1/0:x<0?0.0:1-exp(-lambda*x)" GPFUN_cnormal = "cnormal(x,mu,sigma)=sigma<=0?1/0:0.5+0.5*erf((x-mu)/sigma/sqrt2)" GPFUN_cpareto = "cpareto(x,a,b)=a<=0||b<0||!isint(b)?1/0:x<a?0.0:1.0-(real(a)/x)**b" GPFUN_cpoisson = "cpoisson(x,mu)=mu<=0?1/0:!isint(x)?1/0:x<0?0.0:1-igamma(x+1.0,mu)" GPFUN_crayleigh = "crayleigh(x,lambda)=lambda<=0?1/0:x<0?0.0:1.0-exp(-lambda*x*x)" GPFUN_csine = "csine(x,f,a)=a<=0?1/0: x<0?0.0:x>a?1.0:f==0?real(x)/a:(real(x)/a-sin(f*twopi*x/a)/(f*twopi))/(1.0-sin(twopi*f)/(twopi*f))" GPFUN_ct = "ct(x,nu)=nu<0||!isint(nu)?1/0:0.5+0.5*sgn(x)*(1-ibeta(0.5*nu,0.5,nu/(nu+x*x)))" GPFUN_ctriangular = "ctriangular(x,m,g)=g<=0?1/0: x<m-g?0.0:x>=m+g?1.0:0.5+real(x-m)/g-(x-m)*abs(x-m)/(2.0*g*g)" GPFUN_cuniform = "cuniform(x,a,b)=x<(a<b?a:b)?0.0:x>=(a>b?a:b)?1.0:real(x-a)/(b-a)" GPFUN_cweibull = "cweibull(x,a,lambda)=a<=0||lambda<=0?1/0:x<0?0.0:1.0-exp(-(lambda*x)**a)" GPFUN_rnd = "rnd(x) = floor(x+0.5)" r_xmin = -1 r_sigma = 4.0 n = 10 p = 0.6 mu = 5.33333333333333 sigma = 2.98142396999972 xmin = -1 xmax = 18 ymax = 0.170961981379082 lambda = 0.6 rho = 3.2 nn = 75 mm = 25 r = 8 plot negbin(rnd(x), r, p) with histeps, gmm(x, rho, lambda)