# Countermonotonicity and minimum linear correlation coefficient

In an example exercise they question whether it is possible to construct a bivariate distribution of $LN(0,1)$- and $LN(0,4)$-distributed random variables, where $LN(\mu,\sigma^2)$ is the log normal distribution, such that the linear correlation coefficient $\rho$ is -0.2.

In the answer they say that it can be proven that the minimum corr. coeff. $\rho_{min}$ is given when the two variables are countermonotonic. Thus, $\rho_{min}$ can be found by taking their countermonotonic representation and thereafter it is just straight forward calculations. They find that the minimum correlation coefficient is approximately equal to -0.09. Hence it is not possible to construct such bivariate distribution.

What I wonder is how can I find the countermonotonic representation?

The countermonotonic representation of the log normal r.v.'s is given by $(e^{-Z},e^{2Z})$, where Z is standard normal. But they do not explicitly explain why that is.

I think you can represent the countermonotonic of $\log$ normal by $(1-F(x), F(y))$; $1-F(x)$ implies after the calculations $e^{-Z}$ where $Z$ is $N(0,1)$ distributed.