#This code demonstrates use of rejection sampling to generate a
#random sample from the density that is a half circle on (-1,1)
#i.e., density is (propotional to) f(y)=sqrt(1-y^2)

#Plot the shape of the density
x=seq(-1,1,0.001)
f=sqrt(1-x^2)
par(mfrow=c(2,1),mar=c(5,5,4,4))
plot(x,f,type="l",ylab=expression(sqrt(1-x^2)))

#Generate from Uniform(-1,1)
nsamps=1000000
g=runif(nsamps,-1,1)
hist(g,nclass=100)

#Apply rejection sampling
y=g[rbinom(nsamps,1,sqrt(1-g^2))==1]
hist(y,nclass=100)