Joint distribution from marginals (unknown conditional) distributions

#1
Hi,
This is my first post in this forum; I hope someone can help me with this --thanks in advance for reading!

I'm trying to find the joint probability distribution of two variables where I only know their marginal probability distributions.

More specifically, I'm interested in the joint distribution of two random variables X and Y, given that X is normally distributed (with mean Mx and deviation Sa), and Y= exp[X] (i.e. a log-normal distribution).

I hope it makes sense.

Thanks!
 
#2
If Y = exp(X), that is, Y is strictly a transformation of X, then talking about the "joint distribution" doesn't make a whole lot of sense. OTOH, if you mean that Y is a log-normal random variable which is not strictly a transformation of X, I'd define the variable Z = log(Y) so that Z is a normal random variable, then define the mean vector

[TEX]\mu = (\mu_X, \mu_Z)[/TEX]

and the covariance matrix

[TEX]\Sigma = \text{Cov}(X, Z)[/TEX]

so X and Z have the multivariate normal [TEX](\mu, \Sigma)[/TEX] distribution. The parameters can then be easily estimated from a sample.