Variance Reduction calculate

Question

If $\phi(x)=\frac{e^x-1}{e-1}I_{[0,1]}(x)$, use the variance reduction techniques: Importance Sampling, Antithetic Variables, Control Variates.Compare the methods and check which provides the greatest reduction.

First of all I take $f(x)=1I_{[0,1]}(x)$ $X$~$U[0,1]$. Starting with Antithetic Variables.

$X$~$U(0,1)$ and $Y=1-X$, so the estimator is $$\frac{1}{2n}\sum[\phi(X_i)+\phi(Y_i)]$$

How do I calculate the variance in this case?

Answer 1

You need some basic properties of variance:

https://en.wikipedia.org/wiki/Variance#Basic_properties

and you need to be able to work out $\text{Var}(\exp(X))$ and $\text{Cov}(\exp(X),\exp(1-X))$.

For more details, you need (as the self-study tag-wiki explains) to show a proper attempt and explain where you run into difficulties.