There is much more information in a joint distribution than can
be captured by its marginal distributions.
It is one thing to be told that a joint distribution can't be
constructed from marginals in a unique way. It
is another to have some examples. Here are a few.
Discrete distributions. Consider four different joint distributions
with the same marginals. In all cases, let the marginals have the
distribution Bin(3. 1/2).
Positively Correlated
x/y o 1 2 3 Tot
-------------------------------------
0 1/8 0 0 0 1/8
1 0 2/8 1/8 0 3/8
2 0 1/8 2/8 0 3/8
3 0 0 0 1/8 1/8
-------------------------------------
Tot 1/8 3/8 3/8 1/8 1
Negatively Correlated
x/y o 1 2 3 Tot
-------------------------------------
0 0 0 0 1/8 1/8
1 0 1/8 2/8 0 3/8
2 0 2/8 1/8 0 3/8
3 1/8 0 0 0 1/8
-------------------------------------
Tot 1/8 3/8 3/8 1/8 1
A perfectly correlated example arises from putting the
numbers 1/8, 3/8, 3/8, 1/8 down the main diagonal.
And, of course, there is the independent case in which
the cells are filled by multiplying the marginals.
There are many more examples of different joint distributions that have these same
marginal distributions. And maybe you should try to construct one.
Fill in the body of the table any way you like, using
numbers between 0 and 1 such that the marginal totals
remain unchanged.
Continuous distributions. A slightly more advanced situation comes from the family of
bivariate normal distributions with both means 0 and
both standard deviations 1. And the correlation $\rho$
can take any value between -1 and +1. In this example
both marginal distributions are standard normal, no matter
what the value of $\rho.$
By independence, the joint density function is $f_{X,Y}(x,y) = f_X(x) f_Y(y)$. Be sure to incorporate the information about the support of $f_X$ and $f_Y$ (where it is zero and where it is nonzero).
Similarly, the joint distribution function is $P(X\le x, Y \le y) = P(X \le x) P(Y \le y)$ by independence. Again, be careful: $P(X\le x)$ is piecewise linear.
Best Answer
For example, suppose the marginal densities for $X$ and $Y$ are both 1 on the interval $[0,1]$, 0 otherwise. One family of possibilities for the joint density is $f(x,y) = 1 + g(x) h(y)$ for $0 < x < 1$, $0 < y < 1$, 0 otherwise, for functions $g$ and $h$ such that $\int_0^1 g(x)\, dx = \int_0^1 h(y)\, dy = 0$, $-1 \le g(x) \le 1$ and $-1 \le h(y) \le 1$. And there are infinitely many other possibilities.