Let $X_1,X_2,...,X_n$ be i.i.d. $Exp(\lambda)$ random variables and $Y_k =\sum^{k}_{i=1}X_i$, $k = 1,2,...,n$.
a) Find the joint PDF of $Y_1,...,Y_n$.
b) Find the conditional PDF of $Y_k$ conditioned on $Y_1,....,Y_{k−1}$, for $k = 2,3,...,n$.
c) Show that $Y_1,...,Y_k$ conditioned on $Y_{k+1},...,Y_n$ is uniformly distributed over a subset in $\Bbb{R}^k$, for $k = 1,2,...,n−1$. Find this subset.
My attempt:
For $\lambda_i = \lambda$, $\sum^{n}_{i=1}X_i \sim Erlang(n,\lambda) $, thus $Y_k\sim Erlang(k,\lambda)$
From here I need to find the CDF first to find the PDF. But I don't understand how.
Begin here:
Since for all $2 \leq k\leq n$ we have $X_k=Y_k-Y_{k-1}$, and the $(X_k)$ are iid expnentially distributed with pdf $f_{\small X}(x)=\lambda\exp(-\lambda x)\cdotp\mathbf 1_{0\leq x}$ ... therefore... $$\begin{align}f_{\small Y_1,Y_2,\ldots,Y_n}(y_1,y_2,\ldots, y_n)&=f_{\small X_1,X_2,\ldots,X_n}(y_1,y_2{-}y_1,\ldots,y_n{-}y_{n-1})\\[1ex]&= f_{\small X}(y_1)\prod_{k=2}^nf_{\small X}(y_k-y_{k-1})\\[2ex]&=\lambda^n\exp\left(-\lambda\left(y_1+\sum_{k=2}^n(y_k-y_{k-1})\right)\right)\cdot\mathbf 1_{0\leq y_1\leq y_2\leq\ldots\leq y_n}\\[2ex]&=\phantom{\lambda^n\exp(-\lambda y_n)\cdot\mathbf 1_{0\leq y_1\leq y_2\leq\ldots\leq y_n}}\end{align}$$