Question

5. For α > 0 and β > 0, consider the following accept-reject algorithm: (1) Generate Ul and U2 iid uniform (0. 1) random variables. Set Vi-Ulla and U11/g ; else go to step l (3) Deliver X. Show that X has a beta distribution with parameter α and β.

Answer 1

I have done this problem using R

> rm(list=ls())
> #Generating uniform for Uniform distribution
> u1<-runif(10000,0,1)
> u2<-runif(10000,0,1)
> #now taking alpha(a)=1.2 and beta(b)=1.5
> a<-1.2
> b<-1.5
> #Generating V1 and V2
> v1<-u1^(1/a)
> v2<-u2^(1/b)
> #Generating W
> w=v1+v2
> #Accessing the positions for which w<=1
> pos<-which(w<=1)
> #Accesing v1 and w for which w<=1
> nv1<-v1[pos]
> nw<-w[pos]
> #Prnting how many points under the condition w<=1
> length(pos)
[1] 3532
> #creating Random variable X=V1/W
> x<-(nv1/nw)
> #Plotting the distribution of obtained x
> par(mfrow=c(2,1))
> plot(density(x))

> #Plotting density for beta with parameter alpha=1.2 and beta=1.5 with same
> #length of x
> plot(density(rbeta(length(pos),1.2,1.5)))

> # A view of the generated data.

density.default(x x) 0.0 0.2 0.4 0.6 0.8 1.0 N 3532 Bandwidth 0.04522 density.default(x = rbeta(length(pos), 1.2, 1.5)) 凸丈 0.0 0.2 0.4 0.6 0.8 1.0 N 3532 Bandwidth 0.04495

> head(data.table(u1,u2,v1,v2,w))
u1 u2 v1 v2 w
1: 0.5777117 0.04967083 0.6330325 0.1351246 0.7681571
2: 0.9039229 0.52912337 0.9192694 0.6541911 1.5734605
3: 0.3109522 0.43463077 0.3777850 0.5737821 0.9515671
4: 0.1067732 0.13919905 0.1550191 0.2685906 0.4236098
5: 0.9949222 0.50549031 0.9957667 0.6345637 1.6303304
6: 0.1682814 0.04683880 0.2264801 0.1299383 0.3564184
> head(data.table(nv1,nw))
nv1 nw
1: 0.6330325 0.7681571
2: 0.3777850 0.9515671
3: 0.1550191 0.4236098
4: 0.2264801 0.3564184
5: 0.2086407 0.8312930
6: 0.5256138 0.8683544
> a
[1] 1.2
> b
[1] 1.5
> head(x)
[1] 0.8240925 0.3970135 0.3659480 0.6354333 0.2509834 0.6052988

Conclusion : Density curve obtained by both generated through acceptance rejection algorthim and direct by rbeta() for the parameters alpha=1.2 and beta=1.5 are same hence we can say that x follows beta(alpha,beta)