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This is my first post in one of these 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 non-independent variables given only their marginal distributions. In fact, I'm interested in the joint distribution of two random variables X and Y, both of which are related to transformations of the same (normally distributed) random variable Z. More specifically, I want f(X,Y)=Pr[X=x,Y=y], given that I know:

• X=Z+u

• Y=exp(Z)

• Where Z~N(0,1), u~N(0,1) and COV(Z,u)=0

I know the problem would be trivial if Y was a linear transformation, because the joint would simply be a multivariate normal distribution (i.e. N~(0,Ʃ), where the covariance matrix Ʃ could be expressed in terms of VAR[Z]). In my case the problem is trickier because it involves the joint of a normal and a log-normal distribution.

I hope it makes sense. Any help or hints (also to express the problem more clearly) are very much appreciated.

Thanks!

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# Joint probability distr. of 2 non-independent variables given only marginal distr.

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