Filling out a Joint Distribution Tables Given the marginal distributions. Hello I am trying to figure out how to fill out a joint distribution table when I am only given the marginal distributions.
$$
\begin{array}{c|ccc|l}
_{\large X}\backslash^{\large Y} & 0 & 1 & 2 \\
\hline
0 & \text{?} & \text{?} & \text{?} & 0.5 \\
1 & \text{?} & \text{?} & \text{?} & 0.25  \\
2 & \text{?} & \text{?} & \text{?} & 0.25  \\
\hline
& 0.5 & 0.25 & 0.25
\end{array}
$$
Basically I don't need answers as much as the marginal distributions relate to finding the unknown numbers.

$$
\begin{array}{c|ccc|l}
_{\large X}\backslash^{\large Y} & 0 & 1 & 2 \\
\hline
0 & a & b & (0.5-a-b) & 0.5 \\
1 & c & d & (0.25-c-d) & 0.25  \\
2 & (0.5-a-c) & (0.25-b-d) & (a+b+c+d-0.5) & 0.25 \\
\hline
& 0.5 & 0.25 & 0.25
\end{array}
$$
If we fill in the upper left four cells with $a,b,c,d$, then (since the first row must add up to $0.5$) the third entry in the first row must by $0.5-a-b$, and the other entries are filled in in the same way.
You can put any numbers in place of $a,b,c,d$ that make all entries $\ge 0$ but $\le 1$, and you'll have a probability distribution.
Note that $a$ could be as big as $0.5$, but not bigger, and the table would look like this:
$$
\begin{array}{c|ccc|l}
_{\large X}\backslash^{\large Y} & 0 & 1 & 2 \\
\hline
0 & 0.5 & 0 & 0 & 0.5 \\
1 & 0 & d & (0.25-d) & 0.25  \\
2 & 0 & (0.25-d) & d & 0.25 \\
\hline
& 0.5 & 0.25 & 0.25
\end{array}
$$
and in this case $d$ could be as small as $0$ or as big as $0.25$
[ to be continued, maybe $\ldots$ ]
