# Calculating variance of MC estimator of an integral

Let $$f(x) = \sqrt{1-x^2}, \quad I = \int \limits_0^1 f(x) dx.$$

We consider the following estimator of I: $$\hat{I}_n = \frac{1}{n} \sum \limits_{i=1}^n f(X_i),$$ where $$X_1, X_2, ...$$ are iid $$U(0,1)$$.

The task is to calculate $$Var(\hat{I}_n)$$.

My approach: $$Var(f(X_1)) = E[f(X_1)]^2 - (Ef(X_1))^2 = \int_0^1 1 - x^2 dx - \Big{(} \int_0^1 \sqrt{1 - x^2} dx \Big{)}^2 = \frac{2}{3} - \Big{(}\frac{\pi}{4}\Big{)}^2$$ $$Var(\hat{I}_n) = Var\Big{(} \frac{1}{n} \sum \limits_{i=1}^n f(X_i) \Big{)} = \frac{1}{n^2} Var\Big{(}\sum \limits_{i=1}^n f(X_i) \Big{)} = \frac{1}{n^2} \sum \limits_{i=1}^n Var(f(X_i)) = \frac{1}{n^2} n \Big{(}\frac{2}{3} - \Big{(}\frac{\pi}{4}\Big{)}^2 \Big{)}$$

Is that correct?

My other question is how to estimate $$Var(\hat{I}_n)$$ using simulation?

n <- 100000
U <- runif(n)
X <- sqrt(1-U^2)
I <- mean(X)


Here I is the estimator of the itegral and var(X) gives me approximately $$\frac{2}{3} - \Big{(}\frac{\pi}{4}\Big{)}^2$$

But when I try to estimate $$Var(I_n)$$

w <-c ()
for(i in 1:1000){
U <- runif(n)
X <- sqrt(1-U^2)
I <- mean(X)
w <- c(w,I)
}


Here, var(w) is approximately 5.01359e-07. If I use the formula devired above $$Var(\hat{I}_n) = \frac{1}{n} \Big{(}\frac{2}{3} - \Big{(}\frac{\pi}{4}\Big{)}^2 \Big{)} \approx 4.981639e-05$$

Where is the mistake?

Your calculations seem correct. Since $$n$$ is large, and iterate only 1000 times in the outer loop, the variance you get is fluctuating a lot around small numbers. Try the following, where the variance is better noticeable:

n <- 10
w <-c ()
for(i in 1:50000){
U <- runif(n)
X <- sqrt(1-U^2)
I <- mean(X)
w <- c(w,I)
}
var(w)
(2/3 - (pi/4)^2)/n

• I see :) great :D Thank you very much! – Elizabeth_Banks Jun 5 '20 at 16:44
• It's also quite interesting that in order to calculate the variance of your estimator, we actually use the result of the integral. Then, what was the point in estimating it :) – gunes Jun 5 '20 at 16:45
• That's true :D I did not notice that! The sad answer is: in order to prepare to the exam :) – Elizabeth_Banks Jun 5 '20 at 16:51