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Suppose $(X_1, X_2, X_3, X_4) \sim \text{Dir} (\alpha_1, \alpha_2, \alpha_3, \alpha_4 ; \alpha_5)$, the Dirichlet distribution.

What is the marginal PDF of $(X_i, X_j) \ ; \ i \neq j$?

I have no idea how to do this one.
Please help.

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Hint

Note that

$$ (X_1,X_2,X_3,X_4)\stackrel{d}{=}\frac{(Z_1,Z_2,Z_3,Z_4)}{Z_1+Z_2+\dotsb+Z_5} $$ where $Z_i$ are independent $\text{Gamma}(\alpha_i,1)$ random variables. In particular for example $$ (X_1,X_2)\stackrel{d}{=}\frac{(Z_1,Z_2)}{Z_1+Z_2+(Z_3+\dotsb+Z_5)} $$ sot that $(X_1,X_2)\sim\text{Dir}(\alpha_1,\alpha_2;\alpha_3+\alpha_4+\alpha_5)$. This should help you motivate the general solution supposing that you know the pdf corresponding to a Dirichlet Distribution.

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  • $\begingroup$ What do you mean by $\stackrel{d}{=}$? $\endgroup$ – Alex Mar 19 '18 at 17:40
  • $\begingroup$ Equal in Distribution $\endgroup$ – Foobaz John Mar 19 '18 at 17:41

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