uniqueness of joint probability mass function given the marginals and the covariance

Question

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.

Answer 1

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 $a$.

Answer 2

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.