Joint probability density functions that generate a conservative vector field

Question

Let $X:\Omega\rightarrow[0,1]$ and $Y:\Omega\rightarrow[0,1]$ be random variables and $f:[0,1]^2\rightarrow\mathbb{R}$ be the joint probability density for $X$ and $Y$. I'm interested in the vector field $F:\mathbb{R}^2\rightarrow\mathbb{R}^2$ given by $$F(x,y)=(M(x,y),N(x,y))=(x\cdot f_{X|Y}(x|y),y\cdot f_{Y|X}(y|x))$$ In particular, I'm interested in constraints on $f$ that guarantee that $F$ is a conservative vector field. It is easy to verify that if $X$ and $Y$ are independent, then $F$ is conservative. In that case, $$M(x,y)=x\cdot f_{X|Y}(x|y)=x\cdot f_{X}(x)$$ and likewise $$N(x,y)=y\cdot f_{Y|X}(y|x)=y\cdot f_{Y}(y)$$ Hence $$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}=0$$ What I would like are examples of joint densities $f$ that render $X$ and $Y$ dependent but nevertheless yield $$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$$ so that $F$ is conservative. Better yet, I would like a strategy for identifying constraints on $f$ that are necessary and sufficient for rendering $F$ conservative. Any advice pointing me in the right direction would be much appreciated!

Answer 1

Check perfectly correlated variables, i.e. corr(X, Y) = 1 or -1.