Question

Really short question! Please help me to solve! It also needs the R code, thank you!

(25%)Q2: Please use the accept-reject method to generate 5000 samples from f(x) = 02 n which gfx) - 3e-3) and M f(x) M.g() and prove thet in which gr)3e 1-9(3) 3V2T (1-(3)

Requirements:

Answer 1

Algorithm: 1. Generate zo from the proposal density g 2. Calculate M and f(xo)/(Mg(xo)) 3. Generate u from Uniform(0,1), independently of ro. 4. If u S f(xo)/(Mg(xo)) accept o as a sample from the target density f, otherwise repeat the procedure by generating another ro from the proposal g Proof: Suppose a2 3; otherwise the ratio f(x)/gx) will not be defined. Then we get exp(-9) for constant e. Then log(h(x)) log(c)/2 +3r and3. Thus over the range 2 3, h(x) is decreasing in x and hence maximum occurs at the minimum possible value of x, that is , at x-3. Hence Now Maz.,h(z) = h(3) = cezp(-9/2 + 9) =-φ(3)) lently f(x) Mg(x). -M. Thus h( ) M or equiva-

R Code

n=5000
f=function()
{
M=exp(-4.5)/(3*sqrt(2*pi)*(1-pnorm(3)))
y=numeric(n)
for(i in 1:n)
{
repeat
{
x=3+rexp(1,rate=3)
r=(dnorm(x)/(1-pnorm(3)))/(M*3*exp(-3*x+9))

u=runif(1,0,1)
if(u<=r) break
}
y[i]=x
}
return(y)
}