Let $X$ be a standard uniform random variable, define $Y=1-X$. Then supposedly $X$ and $Y$ are uniform over a 1-simplex, so their joint distribution should be Dirichlet of order $K=2$, and $\alpha_1=\alpha_2=1$ . Hence, their joint pdf is $$f(x,y)=1\quad\text{if }x,y\in[0,1]\land x+y=1.$$ My question is, how do I derive the joint pdf from definition? A naive attempt is to notice the joint cdf $$F(x,y)=P(X\leq x,1-X\leq y)=P(1-y\leq X\leq x).$$ But then taking cross derivatives of $F$ seems yield $0$ everywhere...

  • $\begingroup$ Not everywhere, just almost everywhere. The derivative is not zero along the line $x+y=1$. $\endgroup$ – Paul Jun 7 '16 at 12:09
  • $\begingroup$ @Paul: can you show how it is done? $\endgroup$ – Francis Jun 7 '16 at 12:34
  • $\begingroup$ How do you define a density on $[0,1]^2$? What makes you think that this example has one? $\endgroup$ – Henry Jun 7 '16 at 13:54
  • $\begingroup$ @Henry: I thought there is a pdf defined because the setup looks a lot like a flat Dirichlet distribution of order $2$, and numerical experiment does seem to confirm this. Are you saying that they are in fact different things? $\endgroup$ – Francis Jun 7 '16 at 19:55

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