# set terminal pngcairo transparent enhanced font "arial,10" fontscale 1.0 size 600, 400 # set output 'prob2.5.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 "μ" at 3.33333, 0.0304860, 0.00000 left norotate back nopoint set label 2 "σ" at 4.73046, 0.227608, 0.00000 left norotate back nopoint set arrow 1 from 3.33333, 0.00000, 0.00000 to 3.33333, 0.260123, 0.00000 nohead back nofilled linewidth 1.000 dashtype 3 set arrow 2 from 3.33333, 0.227608, 0.00000 to 4.73046, 0.227608, 0.00000 nohead back nofilled linewidth 1.000 dashtype 3 set samples 13, 13 set style data lines set ytics norangelimit 0.00000,0.030486 ,0.304860 set title "hypergeometric PDF using binomial approximation" set trange [ * : * ] noreverse nowriteback set urange [ * : * ] noreverse nowriteback set vrange [ * : * ] noreverse nowriteback set xlabel "k" set xrange [ -2.00000 : 10.0000 ] noreverse nowriteback set x2range [ * : * ] noreverse writeback set ylabel "probability density" set yrange [ 0.00000 : 0.304860 ] noreverse nowriteback set y2range [ * : * ] noreverse writeback set zrange [ * : * ] noreverse writeback set cbrange [ * : * ] noreverse writeback set rrange [ * : * ] noreverse writeback set colorbox vertical origin screen 0.9, 0.2 size screen 0.05, 0.6 front noinvert bdefault 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) NO_ANIMATION = 1 save_encoding = "utf8" 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 = 0.333333333333333 mu = 3.33333333333333 sigma = 1.39712274047485 xmin = -1 xmax = 9 ymax = 0.30485951131839 lambda = 0.3 rho = 0.7 nn = 75 mm = 25 plot hypgeo(x, nn, mm, n) with histeps, binom(x, n, p) with histeps