Variance Reduction Using Antithetic Variates

Question

I found this online: http://en.wikipedia.org/wiki/Antithetic_variates

For example #2, can someone please provide step by step procedure on how to answer the integral using antithetic variates?

I know I need to produce a random sample from uniform distribution but after that not sure what is next.

Thanks!

Answer 1

For example 2, we are aiming to find an approximation for $I= \int_0^{1} \frac{1}{1+x}dx$. To this aim we will do the following:

Let $U$ be a uniform distribution defined as $U \sim [0,1]$. Then the we have the pdf of $U$, denoted $g$ , is defined as follows: $$ g(x)=\begin{cases} 1 \quad x \in [0,1] \\ 0 \qquad elsewhere \end{cases} $$ Thus given $f(x)= \frac{1}{1+x}$, then $$E[f(U)]=\int_0^{1} f(x) g(x)dx=\int_0^{1} \frac{1}{1+x} dx $$ Thus using this above equality we are able to estimate $I$ by estimating the expectation. Taking a sample $\{ x_i\}_{i=1,..,n}$ of $n$ points in $[0,1]$, then $I$ can be estimated by $$ I \approx \frac{1}{n} \sum_{i=1}^n \frac{1}{1+x_i} $$