#A non-conjugate prior kernel for normal mean:
gfun <- function(mu){
if(mu<=0) return(0)
if(0<mu && mu<3) return(mu)
if(3<=mu && mu<5) return(3)
if(5<=mu && mu<8) return(8-mu)
return(0)}

#Compute the prior density:
prior <- function(mu){
m=length(mu)
value <- c()
for(i in 1:m){
value[i]=gfun(mu[i])}
return(value)}

#Input data:
x=c(4.09,4.68,1.87,2.62)

xbar=mean(x)
n=length(x)

#Compute the likelihood function:
lik <- function(mu){
return(dnorm(xbar,mu,2/sqrt(n)))
}

#Compute the posterior density kernel:
fn1 <- function(mu){return(prior(mu)*lik(mu))}

#Compute the marginal:
mx=integrate(fn,0,8)$value

#Compute the posterior density:
post <- function(mu){return(fn1(mu)/mx)}

#Plot the posterior density:
mu=seq(0,8,l=100)
plot(mu, post(mu), type="l")

#Compute posterior mean:
fn2 <- function(mu){return(mu*post(mu))}
post.mean=integrate(fn2,0,8)$value
cat(c("posterior mean:",round(post.mean,4)),fill=T)

#Compute posterior variance:
fn3 <- function(mu){return(mu*mu*post(mu))}
post.var=integrate(fn3,0,8)$value-post.mean^2
cat(c("posterior variance:",round(post.var,4)),fill=T)
cat(c("posterior sd:",round(sqrt(post.var),4)),fill=T)