conditional normal distribution

Assume two random variables X and Y are not independent,

if P(X), P(Y) and P(Y|X) are all normal, then does P(X|Y) also can only be normal or not necessarily?

thanks.

Hey simon999.

Did you ask this question on physics forums? If so I gave an answer there.

Hi yes I did, not sure I understood your replies though sorry,

so is your answer that P(X|Y) does also has to be normal or not necessarily?

No it has to be if P(Y|X) is proportional to P(X|Y)*P(Y) and if P(X|Y) and P(Y) are both Normal then P(X|Y) also has to be normal.

For a more indepth explanation look up Bayesian statistics with Normal Likelihood and a Normal conjugate prior to see the details.

Are you sure that by having both marginals normal already cover your condition of P(Y|X) is proportional to P(X|Y)*P(Y)

If you want to prove it, take two PDFs that are Normally distributed f(X) and g(X) and then show that f(X)*g(X) also has a normal distribution. This is a very simple exercise and would do you well to understand what is going on.

Thanks,

Someone else tells me


Look at the 2-d density function for dependent normal variables. You will see that, except for singular cases, that the form of the integrand is symmetric in x and y (except for constants). Therefore if one of the conditional distributions is normal, the other must be.

Singular case - correlation = 1.

Is he wrong then?

You can have a variable covariance matrix in the general situation (can you can include this in your PDF's for X and Y if you want).

Sorry so what is the right answer?

if P(X), P(Y) and P(Y|X) are all normal, then does P(X|Y) also can only be normal or not necessarily?

You are saying no but the other person is saying yes