Is the function $F(x,y)=1−e^{−xy}$ $0 ≤ x$, $y < ∞$, the joint cumulative distribution function of some pair of random variables?

How do I show this? I am confused about cumulative distribution functions.

My thoughts so far: $F(x,y)=1−e^{−xy}$ $F(x,y)=1−\frac{1}{e^{xy}}$ so when x or y=0, F(x,y)=0, when x>0 and y>0, F(x,y)=1, and when y<0, $F(x,y)=1−e^{xy}$ so F(x)=0?

I know the joint cdf is related to the joint pdf, maybe I have to do something with that?

  • $\begingroup$ ${\partial ^2 F \over \partial x \partial y}$ would the joint distribution function of your pair of random variables. Integrate this in $\Bbb{R}^2$. If you don't get 1 as a result, then ${\partial ^2 F \over \partial x \partial y}$ is not a joint distribution and hence $F$ is not the cumulative distribution. Note that if you do get 1, then this doesn't tell you anything. $\endgroup$ – Reveillark Mar 3 '15 at 3:11

Hint: The marginal CDFs $F_X(x)$ and $F_Y(y)$ are related to the joint CDF $F_{X,Y}(x,y)$ as follows:
$$F_X(x) = \lim_{y\to \infty} F_{X,Y}(x,y), \quad F_Y(y) = \lim_{x\to \infty} F_{X,Y}(x,y).$$ Apply this to the purported joint CDF to get the (purported) marginal CDFs of $X$ and $Y$. Are these valid CDFs? If so, what do these CDFs suggest about $X$ and $Y$? Is this consistent with what appears to be the joint continuity of $X$ and $Y$?


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