Does Marginal Normality Ensure a Bivariate Normal Distribution?

[learner928]

Hi Viraltux,

the more I think about it, it does seem the only way of keeping marginal distribution of Y normal is for the normal conditional distributions of Y for each value of X follow the bivariate normal formula, with the mean shifting slightly and keeping variance the same.
As soon as you change the variance of these conditional distributions or shift the mean in a different way, the resultant marginal distribution of Y is no longer normal which would break our condition.

do you think so?

 

[Dickfore]

Hey, OP, if the joint probability density function is $\phi(x, y)$, then, how would you express the conditional probability for x, assuming $Y = y_{0}$?

 

[viraltux]

Hi Viraltux,

the more I think about it, it does seem the only way of keeping marginal distribution of Y normal is for the normal conditional distributions of Y for each value of X follow the bivariate normal formula, with the mean shifting slightly and keeping variance the same.
As soon as you change the variance of these conditional distributions or shift the mean in a different way, the resultant marginal distribution of Y is no longer normal which would break our condition.

do you think so?

Maybe... or Maybe you can figure out a way to change the variance through the axis in a way that keeps the marginal normal. For instance, increasing the variance is going to create kurtosis in the marginals as we previously discussed, but how about if you increase the variance for a while and then you reduce it back in a way to amount for the increased kurtosis, would you get a normal marginal? So yeah, until I don't see a formal prove I'll keep my healthy 'I don't know' flashing