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.
Best Answer
\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.