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(X,Y) uniform on $\{(x,y):0<x<y<1\}$, what are the joint and marginal densities?

f(x,y)=c, 0<x<y<1

, as the area is $\frac{1}{2}$ take c=2
Marginal Densities

\begin{displaymath}f_X(x)=\int_{-\infty}^{\infty}f(x,y)dy=\int_{y=x}^1 2 dy= 2(1-x)\end{displaymath}

\begin{displaymath}f_Y(y)=\int_{-\infty}^{\infty}f(x,y)dx=\int_{0}^{x=y} 2 dy= 2y\end{displaymath}

We remarked that :

\begin{displaymath}f_X(x)f_Y(y) \neq f(x,y)\end{displaymath}

Here is what this means:

Susan Holmes