# Antithetic sampling and Monte Carlo simulation

Consider:

\begin{align} f(x) = \left\{ \begin{array}{ll} 0, & 0 < x < 0.9 \\ 100, & 0.9 < x < 0.91 \\ 0, & \textrm{otherwise} \\ \end{array}\right. \end{align}

Determine whether antithetic sampling is helpful, harmful, or neutral for the example above, when evaluating $\int ^1 _0$ f(x) dx through MC simulation.You may do this by finding the variance of the estimator under iid and under antithetic sampling using the same sample size. You may find the variances either theoretically or from a large enough simulation.

Explain your findings from the part above, in terms of the even and odd parts of f.

Don't really know how to proceed. Have calculated E(X) and Var(X) by assuming (applicable for continious functions):

\begin{align} E(X) & = \int ^ \infty _{-\infty} x f(x) dx & Var(X) & = \int ^ \infty _{-\infty} \left(x-\mu\right)^2 dx. \end{align}

I also thought of using MATLAB to create a large sample size with a Monte Carlo simulation, but also a bit unsure of how to proceed.