Suppose that $X$ is a Bernoulli random variable with parameter $p$, and we know that if $2p > 1$, then the antithetic variable of $X$ is $Y = 1 − XZ$, where $Z$ is a Bernoulli random variable with parameter $(1-p)/p$, independent of $X$. What is the antithetic variable when $2p<1$?
Antithetic variate method for Bernoulli random variables
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0What do you call "**the** antithetic variable of $X$"? In the [antithetic variates method](https://en.wikipedia.org/wiki/Antithetic_variates), the second sample is not unique. // If $X$ is Bernoulli with parameter $p<\frac12$ then $Y=(1-X)Z$ is also Bernoulli with parameter $p$ if $Z$ is independent of $X$ and Bernoulli with parameter $p/(1-p)$. – 2017-01-14
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0Yes, you are right. Thanks. – 2017-01-14