[R] Simple MC integration question

[Kehl Dániel]

Dear Community,

I try to compute the variance of a MC integral where I know the 
analytical solution.
The function is exp(-x) integrated on the interval (2,4). (Rizzo example)
The true value is exp(-2)-exp(-4)~.1170196

I have the following short code:
<code>
iter <- 10000
MCs <- numeric(iter)
n <- 5000
a <- 2; b <- 4

for (i in 1:iter){
   x <- runif(n, a, b)
   MCs[i] <- (b-a)*mean(exp(-x))
   }

hist(MCs, freq=FALSE)
<\code>

Which shows a nice normal distribution around the theoretical value, 
with a small variance.
On the 125. page of the book the author says that the
Var(theta_hat)=(b-a)^2/m*Var(g(X)).

I try to compare the theoretical normal distribution with the simulated 
one, but the two normal distributions are far from each other.
I thought Var(g(X)) was 1, because g(x) is exponential with lambda 1, 
but probably thats not true.

Any help is appreciated, thanks:
Daniel