Probability – Joint Distribution Function from Marginals

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Is it possible to obtain joint distribution function when only the marginal distribution functions of random variables are given and, the random variables are not independent?

If possible, it would be helpful if you could provide how to do it, with an example for a two random variables case. I know how to do it, when the random variables are independent.

Best Answer

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 $\rho.$

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