The MAD statistic of an iid sample $(x_1,\ldots,x_n)$ is defined as the median of the absolute deviation from the median: $$ \text{mad}(x_1,\ldots,x_n)=\text{med}\left\{|x_i-\text{med}(x_1,\ldots,x_n)|;\ i=1,\ldots,n \right\}\,. $$
I wonder if there exist non-trivial (continuous) distributions on the $X_i$'s such that the distribution of $\text{mad}(X_1,\ldots,X_n)$ can be obtained in closed form (cdf or density).
The next level of unknown is the derivation of the joint density of the median and the MAD statistics for an iid sample of size $n$.