Re: a newbie question on joint pdf

Ross Clement (Email address invalid - do not use) wrote:
Jerry Dallal wrote:
ouyang.jie@xxxxxxxxx wrote:
Is it possible to calculate or estimate the joint pdf of two random
variables given the marginal pdfs of them. No dependentness constraint
is assumed. Thx in advance.

Not uniquely, that is, the marginal distributions do not determine the
joint distribution.

I realise that participating in "argument threads" is risky, and the
chance of getting good information out of them is small. But can you
expand on "not uniquely"? Keeping the distributions discrete and simple
to avoid having to try and type integrals in ASCII, if we have two
marginal distributions for random variables A with possible values
{c,d} and another random variable B with possible values {c,d}, then we
must have:

p( A=c ) = p( A=c, B=c ) + p( A=c, B=d )
p( A=d ) = p( A=d, B=c ) + p( A=d, B=d )
p( B=c ) = p( A=c, B=c ) + p( A=c, B=c )
p( B=d ) = p( A=d, B=d ) + p( A=c, B=d )

and finally for any probability distribution:

p( A=c, B=c ) + p( A=c, B=d ) + p( A=d, B=c ) + p( A=d, B=d ) = 1.0

A weaker version of the first constraints are (e.g.)

p( A=c, B=c ) <= p( A=c )

These constraints say "something" about the joint distribution, even if
it's not very much. The original question was whether it was possible
to "calculate or evaluate" the joint PDF, so the answer is still
"nope", but surely the most appropriate answer to the question would be
"no, but [as mentioned elsewhere] the joint distribution must be
consistent with the marginal distributions.

Cheers,

Ross-c


"In the beginning," there is the universe of all possible bivariate distributions.

Specifying the marginal distributions narrows down the field of all possibilities considerably, so you can say something, but are ways away from specifying the actual bivariate distribution.

Let the bivariate distribution be denoted F(X,Y). Let the marginal distributions be denoted F(1*X+0*Y) and F(0*X+1*Y).

Consider the general case F(a*X+(1-a)Y). Then, the usual marginal distributions are two special cases, one for a=0 the other for a=1.

The more distributions you know--that is, the more values of a for which F(ax+(1-a)Y) is known--the more you can narrow down the field. I'll leave the final word to Professor Rubin, but I believe that if you know F(aX+(1-a)Y) for ALL 0<=a<=1) you CAN characterize the bivariate distribution.

So, "not uniquely" is TRUE on its face in that some possibilities for bivariate F(X,Y) can be ruled out by knowing the marginals. I chose non-uniquely because if you think of the marginals as special cases of F(aX+(1-a)Y), you can keep ruling out more and more possibilities, the more values of 'a' you know.

If I'm wrong about F(aX+(1-a)Y) for all 'a' characterizing the bivariate distribution, then I'm happy to leave it as an open question for the more mathematical for the group, Is there *any* way to characterize a bivariate distribution from its one-dimensional projections?
.

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