Consider the following expression
$$D=X-(X+Y)(X+Z) \ \ \text{under the constraint} \ \ X+Y+Z \leq 1 \tag{1}$$
(being understood that $X, Y$ and $Z$ are $\geq 0$).
This expression can also be written under the form :
$$D=XT-YZ=\begin{vmatrix}X&Y\\Z&T\end{vmatrix} \ \text{given that} \ X+Y+Z+T=1.\tag{2}$$
where $(X,Y,Z,T)$ take their values in $[0,1]^4$ ($[0,1]$ being the set of real values between $0$ and $1$).
(all these constraints are due to the fact that we deal with a complete system of events ; see "Origin of the question" below with definitions (4)).
My question : what is the pdf (probability density function) of random variable D ?
My work :
I have attempted different computations, but they have failed to be conclusive ; I hit on the difficulty to assign a distribution to the random vector $(X,Y,Z,T)$ taking into account constraint $X+Y+Z+T=1$. I have in particular attempted to replace uniformly distributed $X,Y,Z,T$ on $\mathbb{R}$ by random variables taking discrete values $\{0/n,1/n,\cdots (n-1)/n,1\}$ without success.
I have made simulations displaying a rather "smooth" symmetrical behavior (see Fig. below displaying an histogram for a large number of simulations) with a relative spike around value $d=0$.
The origin of this question : A recent question about extreme values of what could be called a "degree of dependence':
$$D=P(A \cap B) - P(A)P(B) \tag{3}$$
(that can be transformed into (1)) and the answer I gave : https://math.stackexchange.com/q/3090278 has triggered (as often is the case) a different question : instead of restricting our attention to the extreme values of (1) (or the equivalent expression (2)), I asked myself the more general question of the pdf of the values taken by (1) with the following definitions :
$$X=P(A \cap B), \ Y=P(A \cap B^c), \ Z=P(A^c \cap B), \ T=P(A^c \cap B^c).\tag{4}$$
as one can see globally on the following diagram :
Besides, if one drops constraint $X+Y+Z+T=1$ (yielding a very different issue), what would become the pdf of $D$ ?(I give below a histogram gathering results of simulation with this hypothesis).
