## bivariate normal distribution

Quick question on bivariate normal distribution please:

I know for a bivariate normal distribution, the two variables are marginally normal and all the conditional distributions are also normal.

Is the reverse true?

i.e. if you have two random variables that are marginally normal themselves and all the conditional distributions of one variable given a value of the other are also normal, does this result in the joint distribution has to be "bivariate normal" or it can still be another type of joint distribution?

thanks.

 Quote by learner928 Quick question on bivariate normal distribution please: I know for a bivariate normal distribution, the two variables are marginally normal and all the conditional distributions are also normal. Is the reverse true? i.e. if you have two random variables that are marginally normal themselves and all the conditional distributions of one variable given a value of the other are also normal, does this result in the joint distribution has to be "bivariate normal" or it can still be another type of joint distribution? thanks.
In fact as long as they both are normal (and independent) you are done; the joined distribution will be bivariate normal.

Recognitions:
 Quote by viraltux In fact as long as they both are normal (and independent) you are done; the joined distribution will be bivariate normal.
Independence is unnecessary.

## bivariate normal distribution

 Quote by mathman Independence is unnecessary.
I didn't say it is mandatory, but if it is there then you are sure you have a jointly bivariate normal, if it is not then you have to prove it.
 thanks guys, just to be clear, even if they are not independent, is the joint distribution has to be bivariate normal? no need for proof.

 Quote by learner928 thanks guys, just to be clear, even if they are not independent, is the joint distribution has to be bivariate normal? no need for proof.
Well, as I said, if X and Y are normally distributed but not independent, they might be or might be not jointly normally distributed; you need to prove it per each case.
 on top of X and Y are normally distributed but not independent, if all the conditional distributions are also normal, does that mean X and Y are definitely jointly normal?

 Quote by learner928 on top of X and Y are normally distributed but not independent, if all the conditional distributions are also normal, does that mean X and Y are definitely jointly normal?
I don't think so, the moment you are allowed to create dependencies between X and Y you can always look for a freak relationship that breaks the definition of the bivariate jointly distribution.
 thanks, so you don't think the fact not only all conditional distributions are normal, X and Y are also marginally normal, this is not enough to restrict the joint distribution to be bivariate normal?

 Quote by learner928 thanks, so you don't think the fact not only all conditional distributions are normal, X and Y are also marginally normal, this is not enough to restrict the joint distribution to be bivariate normal?
Nah, I don't think so. The definition of a multivariate normal distribution is not simple, one of the condition it has to follow (among other more complex than this one) is that every linear combination of its components is also normally distributed .

You could try to further constraint X and Y to behave in such a way that a particular linear combination of their values would not behave normally even if for every particular value of X and Y they do (which is basically what your restriction does). And if that does not work you could still try to mess with the other conditions.
 thanks, i don't have the proof, but I think the fact because X and Y are also marginally normal (on top of all conditional distributions are normal), this extra condition does make every linear combination of X and Y normally distributed hence bivariate normal, you don't think that is the case? any idea how to proof?
 Mathman do you know who is correct?

 Quote by learner928 thanks, i don't have the proof, but I think the fact because X and Y are also marginally normal (on top of all conditional distributions are normal), this extra condition does make every linear combination of X and Y normally distributed hence bivariate normal, you don't think that is the case? any idea how to proof?
It really looks like that, does it? Your restrictions enforce a good amount of independence between X and Y and make it difficult to find dependencies between X and Y to break the bivariate normality but, anyway, how about this one:

Imagine that given X=x the variance of the normal conditional distribution of Y is inversely proportional to x, and also imagine that the variance of X condition to Y=y is also inversely proportional to y.

Now you have X,Y, X|Y=y and Y|X=x following normal distributions, but you are getting in the bivariate distribution a contour line that looks nothing like an ellipse which is what you would expect if it would follow a bivariate joint normal distribution... yeah? I let for you the fun to do the formal proof though

Recognitions: