Importance Sampling when sampling from empirical distribution

Assume we work with the statistic mean, and we are interested in upper tail prob.
P(mean(x) > 0.4). The bootstrap population is the empirical distribution, i.e. equally likely,
(the nonparametric bootstrap)


xvec <- rnorm(30)   ### the sample
pvec <- exp(1.3*xvec)
pvec <- pvec/sum(pvec)   ### the tilted prob vec, take lambda =1.3


sample(x=xvec, size=30, replace=TRUE, prob=pvec)           ### no longer equally likely, larger x is more likely

mean(sample(x=xvec, size=30, replace=TRUE, prob=pvec) )  ### should be larger than 0

#### I need to keep track which x-value got resampled, according to which prob.

indexvec <- sample(x=1:30, size=30, replace=TRUE, prob=pvec)
xvec[indexvec]
pvec[indexvec]

OK. Now in a loop to compute average

for(i in 1:1000) {
indexvec <- sample(x=1:30, size=30, replace=TRUE, prob=pvec)
bootvec <- xvec[indexvec]
pvecboot <- pvec[indexvec]
result[i] <-  as.numeric(mean(bootvec) > 0.4) /prod( 30*pvecboot )      ### not very efficient, for those mean( ) <0.4, I do not need the prod( )
}
mean(result[1:1000])   ### (1)


Compare this with the plain Monte Carlo

for(i in 1:1000) {
bootvec <- sample(x=xvec, size=30, replace=TRUE)
result[i] <-  as.numeric(mean(bootvec) > 0.4)     
}
mean(result[1:1000])   ### (2)

The (2) should be more variable than (1).  Verify this by repeat the above many times and plot.