In studying joint distribution and density functions I have become confused by this exponential distrobution:
$$F_{X,Y}(x,y)=(1-e^{-(x+y)})u(x)u(y)$$
It seems this should be a proper distribution as it is:
- monotonically increasing
- $F_{X,Y}(0,0)=F_{X,Y}(-\infty,y)=F_{X,Y}(x,-\infty)=F_{X,Y}(-\infty,-\infty)=0$
- $F_{X,Y}(\infty,\infty)=1$
- $0\le F_{X,Y}(x,y)\le1$
However if I take the partial derivatives with respect to X and Y to derive the density I get a function that is not a valid density:
$$\frac{\partial}{\partial x \partial y} F_{X,Y}(x,y)=\delta(x) \delta(y)-e^{-(x+y)}u(x)u(y)+e^{-(x+y)}u(x)\delta(y)+e^{-(x+y)}\delta(x)u(y)-e^{-(x+y)}\delta(x)\delta(y)$$
Which basically reduces to $-e^{-(x+y)}u(x)u(y)$ for non-zero values of X and Y. But this is not a valid density function. Can anyone offer insight on how I am going astray?