n=10^6
x=rexp(n,1)
u=runif(n,0,1)
x=x[u<1/3*(x-1)^2/(4/3*exp(4)*exp(-x)) & x<=3]
hist(x,breaks=100,freq=F)
xv=seq(0,3,l=1000)
lines(xv,1/3*(xv-1)^2,col="red",lwd=3)
length(x)/n

n=10^6
x=rexp(n,1)
u=runif(n,0,1)
x=x[u<1/2*sin(x)/(4/3*exp(pi)*exp(-x)) & x<=pi]
hist(x,breaks=100,freq=F)
xv=seq(0,pi,l=1000)
lines(xv,1/2*sin(xv),col="red",lwd=3)
length(x)/n


t=1/3
n=10^5
x=runif(n,1,10)
u=runif(n,0,1)
x=x[u<t/(x)^(t+1)]
hist(x,breaks=c(seq(0,10,by=0.1),max(x)+1),freq=F,xlim=c(0,6))
xv=seq(0,6,l=1000)
lines(xv,t/(xv)^(t+1),col="red",lwd=3)
length(x)/n