# Dirichlet Distribution and Beta Distribution

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.

• There's a pattern in the nested integration. – Lee David Chung Lin Sep 29 '18 at 16:43
• Try to generalize the answers here:math.stackexchange.com/questions/543764/…, math.stackexchange.com/questions/1064995/… – StubbornAtom Sep 29 '18 at 17:50
• ?? The proof is exactly the same for $k=3$ and for higher values of $k$. Please show what you did to solve the case $k=3$. – Did Sep 29 '18 at 18:00
• @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 – mathstruggles Sep 29 '18 at 18:28
• 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? – 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.