Monte Carlo simulation of $\hat I=\frac{1}{n}\sum_{i=1}^n xe^{-x}sin(x)$

Question

I'm trying to evaluate $$I=\int_{0}^\pi xe^{-x}sin(x) dx$$

by doing the Monte Carlo simulation in R with the following estimator

$$\hat I=\frac{1}{n}\sum_{i=1}^n xe^{-x}sin(x)$$

where X₁, X₂,...,X_n are i.i.d. continuous random variables of $Unif(0, \pi)$.

When I integrate $I$, I get $I=0.5895$.

Now, when I try to estimate $\hat I$ in R with n=10 000 with the following instructions:

x<-runif(10000, min = 0, max = pi)

mean(x*exp(-x)*sin(x))

$= 0.1876$

Is it normal it is so far off from the true value? Am I missing something when I type it in R?

Thanks in advance.

Answer 1

Hint: multiplying your answer by $\pi$ gives the actual integral's value.

The PDF of the $\text{Unif}(0, \pi)$ distribution is $f(x)=\frac{1}{\pi} \mathbf{1}_{[0,\pi]}(x)$. You forgot to account for this factor of $\frac{1}{\pi}$ when defining your approximation.

Answer 2

Monte Carlo gives the average of the integrand over the interval. To get the integral, multiply the average by the length of the interval.