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.

You can setup a system of linear equations and inequalities that characterizes the set of all feasible solutions. Given $P(X=k),$ $k\ge 0$, $P(Y=h),$ $h\ge 0$ and $E[XY]$ this system will be 
$$
\begin{array}{rcl}
\sum\limits_{h\ge0}g_{h,k}&=&P(X=k),\ k\ge 0,\\
\sum\limits_{k\ge0}g_{h.k}&=&P(Y=h),\ h\ge 0,\\
\sum\limits_{k,h\ge 0}khg_{h,k}&=&E[XY],\\
\sum\limits_{k,h\ge 0}g_{h,k}&=&1,\\
g_{kh}&\ge& 0,\ k,h\ge 0.
\end{array}
$$
The solution is not unique whenever the support set of at least one of the variables has three or more values and the other variable is not constant (support has at least two values). This is because you have $|supp X|+|supp Y|$ independent equations but $|supp X|\times |supp Y|$ unknowns. 
