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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?

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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
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    $\begingroup$ I see :) great :D Thank you very much! $\endgroup$ – Elizabeth_Banks Jun 5 '20 at 16:44
  • $\begingroup$ 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 :) $\endgroup$ – gunes Jun 5 '20 at 16:45
  • $\begingroup$ That's true :D I did not notice that! The sad answer is: in order to prepare to the exam :) $\endgroup$ – Elizabeth_Banks Jun 5 '20 at 16:51

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