Say $f(X_1)$, $f(X_2)$, $f(X_3)$, $f(X_4)$ are the empirical marginal PDFs of random variables $X_1$, $X_2$ , $X_3$, $X_4$. Also given is correlation between each pair of variables $X_1$, $X_2$ , $X_3$, $X_4$. Is it possible to obtain the empirical joint PDF of $f(X_1, X_2, X_3, X_4)$?

  • $\begingroup$ How do you define "empirical marginal PDFs"; $\endgroup$ – JohnK Jul 9 '16 at 17:08
  • $\begingroup$ @JohnK I obtained empirical PDFs from an experiment that I conducted $\endgroup$ – Spandyie Jul 9 '16 at 17:11
  • $\begingroup$ not without additional assumptions (eg. multi-variate normality would be sufficient). $\endgroup$ – Matthew Gunn Jul 9 '16 at 17:28

In general, no.

In fact, even if the marginals are normally distributed, the joint distribution could be quite interesting. Here is an one example: Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian?.

  • $\begingroup$ @Vimal the marginals of individual variables are definitely not normally distributed, I think it looks somewhat Gamma distributed. $\endgroup$ – Spandyie Jul 10 '16 at 1:09

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