Assume X and Y have a bivariate lognormal distribution (x,y>0). I want to know the median of |x-y|.
first I calculate the density of |x-y| based on changing the variables, as follows:
I took u=|x-y| and v=y then
\begin{cases}
x=u+v, & \text{if x$\gt$y which result in: u>0 , v>0} \\
x=v-u, & \text{if x$\lt$y which result in v>u>0}
\end{cases}
the Jacobian is one,
so the density of u=|x-y| is:
$$\int_{0}^\infty f_{X,Y}(u+v,v)dv+\int_{u}^\infty f_{X,Y}(v-u,v)dv$$
my question is: Is this the correct way to estimate the density of absolute value of the difference between two log normal random variables and how can I calculate the median of the above density? appreciate any help