Max(F(x),F(y)) is a joint CDF? If F(•) is a cumulative distribution function:

•) Is F(x,y) =max[F(x),F(y)] a joint cumulative distribution function? 

I intuitively think that it is not a cumulative distribution function

:( I have no idea how to start this exercise, for your help, thank you very much.
Hint:

If $F(x,y)$ is a joint cdf, and $x_1&lt;x_2$ and $y_1&lt;y_2$, then you must have
$$
F(x_2,y_2)-F(x_1,y_2)-F(x_2,y_1)+F(x_1,y_1)\ge 0,
$$
since the LHS is the probability of an event (what event?). Let $y_1=x_1$ and $y_2=x_2$, and show the above is not satisfied in general for your joint cdf. 



Perhaps even simpler; any joint cdf must satisfy
$$
\lim_{x\to-\infty} F(x,y)=0
$$
for any fixed $y$. But $\lim_{x\to-\infty}\max(F(x),F(y))=F(y)$.
We already know that 
$$
\lim_{x\to \infty} F(x)=1.
$$
The thing to note is that to be a joint CDF, we must have
$$
\lim_{x\to\infty, y\to \infty}F(x,y)=1
$$

Here is the proof: we have
$$
\lim_{x\to\infty, y\to \infty}F(x,y)
=\lim_{x\to\infty, y\to \infty}\max(F(x),F(y))\geq\lim_{x\to \infty} F(x)=1,
$$
since $\max(F(x),F(y))\geq F(x)$.

But we also have $F(x)\leq 1$ for all $x$, so $\max(F(x),F(y))\leq 1$.

This means $\lim_{x\to\infty, y\to \infty}F(x,y)=1$.

There are other details to consider. $F(x,y)$ need to be increasing with $x$ and $y$. So, let's suppose that for some $x_1&lt;x_2$, $F(x_1,y)&gt;F(x_2,y)$. A moment's thought could lead to that we must have $F(x_1)\leq F(y)$ and $F(x_2)\geq F(y)$, a contradiction. (I have omitted the details) Actually, in general, the "max" of two increasing functions is increasing.

The conclusion is that $F(x,y)$ is a joint CDF. The same is, however, not true for PDF.