So I've seen questions on here about Dirichlet and integrating it, but they seem to be simple questions. I'm stuck with this:

"Let $Y_1, \ldots , Y_k$ have a Dirichlet distribution with parameters $α_1, \ldots, α_k, α_{k+1}.$

Show that $Y_1$ has a beta distribution with parameters $α = α_1$ and $β = α_2 + \cdots + α_{k+1}$."

Looking around the website I understand the case for when $k$ is small like $3,$ but what do I do with $k+1$? Do I have to integrate constantly until I get down to $Y_1$? Or is there something I'm missing

EDIT: So I have started with this as my pdf for $Y_1,\ldots,Y_k$ -

$$g(y_1, \ldots , y_k) = \frac{Γ(α_1 + \cdots + α_{k+1})}{Γ(α_1) \cdots \Gamma(α_{k+1})} y^{α_1−1}_1\cdots y^{α_k−1}_k (1−y_1−\cdots−y_k)^{α_{k+1}−1}$$

So I want to find $g_{Y_1}(y_1)$, and to do so I'd need to

$$g_{Y_1}(y_1)= \int_{-\infty}^\infty g(y_1, \ldots , y_k) \, dy_2\,dy_3\cdots dy_k \text{?}$$

For the $k=3$ case, I know that you substitute $y=(1-x)v$ and then integrate, get to a step where you have the distribution of $\operatorname{Beta}(α_2,α_3).$

Do I have to integrate and substitute one $y_i$ at a time for $2\leqslant i \leqslant k$? Or is it a one big substitution.

  • 1
    $\begingroup$ There's a pattern in the nested integration. $\endgroup$ – Lee David Chung Lin Sep 29 '18 at 16:43
  • $\begingroup$ Try to generalize the answers here:math.stackexchange.com/questions/543764/…, math.stackexchange.com/questions/1064995/… $\endgroup$ – StubbornAtom Sep 29 '18 at 17:50
  • $\begingroup$ ?? The proof is exactly the same for $k=3$ and for higher values of $k$. Please show what you did to solve the case $k=3$. $\endgroup$ – Did Sep 29 '18 at 18:00
  • $\begingroup$ @Did my work would be the exact same as the link posted by StubbornAtom above. Would I extend the proof by substituting all the Y's as one variable? Or would I have to integrate numerous times $\endgroup$ – mathstruggles Sep 29 '18 at 18:28
  • $\begingroup$ Sorry but this won't do: please show exactly what you did and where you meet problems -- otherwise how do you expect we can help? $\endgroup$ – Did Sep 29 '18 at 18:31

\begin{align} & \int\limits_{\left\{ \begin{array} c (y_2,\,\ldots,\,y_n) \,: \\ y_2+\,\cdots\,+ y_n = 1 \end{array} \right\}} \cdots\cdots \,dy_1 \cdots dy_k \\[12pt] = {} & \int_0^1\left( \int_0^{1-y_k} \left( \int_0^{1-y_k - y_{k-1}} \left( \int_0^{1-y_k-y_{k-1}-y_{k-2}} \cdots\cdots \, dy_{k-3} \right) \, dy_{k-2} \right) \, dy_{k-1} \right) \, dy_k \end{align}

Try this for $k=2,$ then for $k=3,$ then for $k=4,$ and you'll see the pattern.


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