# set terminal pngcairo transparent enhanced font "arial,10" fontscale 1.0 size 600, 400 # set output 'prob2.8.png' 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 12.5000, 0.00847735, 0.00000 left norotate back nopoint set label 2 "sigma" at 17.9772, 0.0441776, 0.00000 left norotate back nopoint set arrow 1 from 12.0000, 0.00000, 0.00000 to 12.0000, 0.0728366, 0.00000 nohead back nofilled linewidth 1.000 dashtype solid set arrow 2 from 12.0000, 0.0441776, 0.00000 to 17.4772, 0.0441776, 0.00000 nohead back nofilled linewidth 1.000 dashtype solid set style increment default set samples 200, 200 set style data lines set ytics norangelimit 0.00000,0.00847735,0.0847735 set title "negative binomial PDF using normal approximation" set xlabel "k, x ->" set xrange [ -2.00000 : 35.0000 ] noreverse nowriteback set x2range [ * : * ] noreverse writeback set ylabel "probability density ->" set yrange [ 0.00000 : 0.0847735 ] noreverse nowriteback set y2range [ * : * ] noreverse writeback set zrange [ * : * ] noreverse writeback set cbrange [ * : * ] noreverse writeback set rrange [ * : * ] noreverse writeback 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||x1?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?0.0: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?0.0:1.0/g-abs(x-m)/real(g*g) uniform(x,a,b)=x<(a=(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||xd||x>C?1.0:hypgeo(x,N,C,d)+chypgeo(x-1,N,C,d) claplace(x,mu,b)=b<=0?1/0:(x1?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:xa?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?1.0:0.5+real(x-m)/g-(x-m)*abs(x-m)/(2.0*g*g) cuniform(x,a,b)=x<(a=(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) fourinvsqrtpi = 2.25675833419103 invpi = 0.318309886183791 invsqrt2pi = 0.398942280401433 log2 = 0.693147180559945 sqrt2 = 1.4142135623731 sqrt2invpi = 0.797884560802865 twopi = 6.28318530717959 r_xmin = -1 r_sigma = 4.0 n = 10 p = 0.4 mu = 12.0 sigma = 5.47722557505166 xmin = -1 xmax = 34 ymax = 0.084773490519756 lambda = 0.6 rho = 3.2 nn = 75 mm = 25 r = 8 plot negbin(rnd(x), r, p) with histeps, normal(x, mu, sigma)