uniqueness of joint probability mass function given the marginals and the covariance Let X and Y be two nonnegative, integer-valued random variables. Is there a way to find the joint probability mass function, i.e.
$$ 
\mathbb{P}(X= k, Y= h) 
$$
for some $k,h\geq 0$, given the marginals and the covariance? 
Moreover, it is true that $\mathbb{E}[XY]=\sum_{k=0}^{\infty}\sum_{h=0}^{\infty}h\,k\,\mathbb{P}(X= k, Y= h)$. 
I know that I can write 
$$
\mathbb{E}[XY]=\sum_{k=0}^{\infty}\sum_{h=0}^{\infty}h\,k\,g_{h,k}
$$
Can I conclude that $\mathbb{P}(X= k, Y= h)=g_{h,k}$? Is there uniqueness?
Thanks in advance.

In general $E[X]=\sum_x xp(x)$; So, obviously, for $Z=XY$ we would have
\begin{align*}
E[Z]=&\sum_z zP(Z)\\
=&\sum_x \sum_y xy P(X,Y)\\
=&~a
\end{align*}
where $a$ is a constant. But your question is how many coefficient sets $\{c_{x,y}\}$ we can find so that
\begin{align*}
\sum_x \sum_y x~y~c_{x,y}=a
\end{align*}
For sure it is not unique. For example for an arbitrary pair ($x_1$ and $y_1$) if you change $c_{x_1,y_1}$ to $c_{x_1,y_1}+1$ then the total result would be increased by $x_1 y_1$. Now, for another arbitrary pair ($x_2$ and $y_2$), you can compensate this difference by decreasing $c_{x_2,y_2}$ value to become $(c_{x_2,y_2}-\frac{x_1 y_1}{x_2 y_2})$.
As you see there exist infinite sets of $\{c_{x,y}\}$ which all of them result in $a$.
