Independence of Beta ratios of Gamma variates If $X= x_1/(x_1+x_2+x_3)$ and $Y= x_2/(x_1+x_2+x_3)$ where $x_1, x_2, x_3$ are independent $\chi^2$-distributed random variables with d.f. $-n_1,n_2, n_3$ respectively. Are $X$ and $Y$ independent?
I am familiar with the usual joint distribution method to check for independence but I am looking for a shorter approach.
 A: The question does not really have anything to do with the Chisquared distribution: it is about the relationship between the new ratios $X$ and $Y$ and whether that structure creates dependency. One can always easily check this oneself by generating some random values for each of the $X_i$, then generate $X$ and $Y$, and then plot $X$ vs $Y$. 
Or better if you wish to think about it ...
With 2 variables:
If $X = \frac{X_1}{X_1+X_2}$ and $Y = \frac{X_2}{X_1+X_2}$, then $X$ and $Y$ are two parts of the same ratio that must sum to 1 (i.e. $X+Y=1$), and the relationship between $X$ and $Y$ could be plotted as:

With 3 variables:
With the 3rd variable $X_3$ added, we have $X = \frac{X_1}{X_1+X_2 + X_3}$ and $Y = \frac{X_2}{X_1+X_2+X_3}$. If we define a third ratio $Z = \frac{X_3}{X_1+X_2 + X_3}$, then $X + Y + Z = 1$  (i.e. that $X + Y <1$) and thus that the domain of support for $X$ and $Y$ could be plotted as the shaded space below the line:

If you were to generate pseudorandom data for $X$ and $Y$ and plot it, it would look something like this:

A: If$$X= x_1/(x_1+x_2+x_3)\quad\text{and}\quad Y= x_2/(x_1+x_2+x_3)\qquad x_i\stackrel{\text{ind}}{\sim}\chi^2_{n_i}\quad i=1,2,3$$then, as shown by whuber
$$(X,Y,1-X-Y)\sim\text{Dir}_3\left(n_1,n_2,n_3\right)$$a Dirichlet distribution with density
$$\frac{\Gamma(n_1+n_2+n_3)}{\Gamma(n_1)\Gamma(n_2)\Gamma(n_3)}x^{n_1-1}y^{n_2-1}(1-x-y)^{n_3-1}$$
Therefore the density of $(X,Y)$ does not separate into a function of $x$ and a function of $y$, ergo, they are dependent. Of course, a simpler and correct argument is that, since $0\le X+Y\le 1$ the supports of $X$ given $Y=y$ is $(0,1-y)$ hence depends on the realised value of $Y$.
