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I have two random variables $\alpha(x)$ and $\beta(x)$, and they are correlated. $\alpha(x)$ obeys log-normal distribution while $\beta(x)$ obeys normal distribution. How do I construct a joint probability distribution function in this case?

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  • $\begingroup$ It totally depend on your correlation $\endgroup$ – Tryss Jun 13 '16 at 12:45
  • $\begingroup$ I just want some general rules to construct the PDF in the case of correlated random variables. I am not concerned about the details of correlation. $\endgroup$ – titanium Jun 13 '16 at 12:52
  • $\begingroup$ But the correlation is fundamental : there is an infinity of such joint probability (and they are very different). $\endgroup$ – Tryss Jun 13 '16 at 13:27
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Hint

let $X=e^{\alpha(x)}$ and $Y=\beta(x)$, now we can say $X$ is a normal random variable.then $${{f}_{X,Y}}(x,y)=\frac{{{e}^{-\frac{1}{2(1-\rho {{*}^{\,2}})}}}}{2\pi {{\sigma }_{1}}{{\sigma }_{2}}\sqrt{1-{{\rho }^{*}}^{\,2}}}\left[ {{\left( \frac{x-{{\mu }_{1}}}{{{\sigma }_{1}}} \right)}^{2}}-2{{\rho }^{*}}\left( \frac{x-{{\mu }_{1}}}{{{\sigma }_{1}}} \right)\left( \frac{y-{{\mu }_{2}}}{{{\sigma }_{2}}} \right)+{{\left( \frac{y-{{\mu }_{2}}}{{{\sigma }_{2}}} \right)}^{2}} \right]$$

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