Quick question on bivariate normal distribution,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 variables that are marginally normal and all the conditional distributions are also normal, is the joint distribution of these two variables has to be "bivariate normal"?Thanks

This is not a proof just an idea. We are asked about joint distribution. We know the formula that expresses joint distribution as the product of conditional probability and unconditional probability of hypothesis. Therefore, to answer the question one needs information about single distribution of the one rv along with correspondent conditional distribution. Thus probably given assumptions are not complete for positive answer. Though it's only a guess.

So given each of the two variables are marginally normal 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?

my idea belong to general setting: whether or not conditional distributions E { Z | X } , E { Z | Y } of 2-dimensional random variable Z = ( X , Y ) are sufficient to state existence and uniqueness of the distribution Z. In general it looks like no. To justify this conclusion it is sufficient to construct an example. Try that it is always helpful. It seems that example should look like X , Y = X + V where V should be chosen in a particular way.Actually normal distribution is a specific one and as always assuming something specific one may prove more than in general case. An example might show because X, Y , V could be normal in particular.

Don't think my query was really answered though.Does anyone know whether,If two non-independent random variables X and Y are both themselves marginally normal, and all conditional distributions (distribution of X for a particular value of Y or vice versa) are also normal, does this mean the joint distribution of X and Y must be bivariate normal or not necessarily so?Thanks.

Here is an irritating technicality which may be relevant: Let X be normally-distributed with mean 0 and variance 1. Let Y = X.Their joint distribution isThis isn't a bivariate normal PDF, but it is the limit of a bivariate normal PDF that's very skinny in one direction. So it looks like a counterexample... but not really. The conditional probabilities aren't normal; they're the limit of a very skinny normal distribution.EDIT: I think the Wikipedia article on joint probability distribution may prove your conjecture. If p(x,y) is the product of f(x) and g(y), and f() and g() are both normal, then p(x,y) has to be a bivariate normal distribution. Anyone see any flaws in this argument?

Both p(y) and p(x|y) are normal. We can write generallyThe joint is given by the product p(y) p(x|y). Now we can find p(y|x) aswhere I only kept terms dependent on y. The question is what functions b and c make this a quadratic function of y. The answer is that b must be a constant and c - linear function of y. The proof is straightforward because the expression is also a quadratic function of x (so you know that b(y) is at most a quadratic function of y, and so is b(y) c(y) and 1/2 ln(b(y)) - 1/2 b(y)c(y)^2).