I am reading a book on antithetic sampling.It is said that the idea of antithetic sampling can be applied when it is possible to find transformation of $X$ that leave its measure unchanged (for example, if $X$ is Gaussian, then $-X$ is Gaussian as well).Suppose that we want to calculate $$ I=\int_0^1{g(x)dx}=E[g(x)] $$ with $X$~$U(0,1)$, and $U$ denotes uniform distribution.The transformation $x\rightarrow 1-x$ leaves the measure unchanged (i.e,$1-X$~$U(0,1)$), and $I$ can be rewritten as $$ I=\frac{1}{2}\int_0^1{(g(x)+g(1-x)})dx. $$ My question is:
Is the second integration only apply for uniformly distributed variables? Or it can be used for variables with any distribution? Thank you!
For any random variable $X$ which is symmetric with respect to $\frac{1}{2}$, $X$ and $1-X$ have the same distribution.