# Conditional and joint distribution of the sum of exponential RVs

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:

By change of variable I got: $$f_{Y_1 ,..., Y_ n} ( y_ 1 ,..., y_ n )=λexp(−λ y_ n )$$

for the b) using: $$f_{ Y_ k | Y_ 1 ,..., Y_{ k−1}} ( y_ k | y_ 1 ,..., y_{ k−1} )= \frac{ f_{ Y_ 1 ,.., Y_ k} ( y_ 1 ,..., y_ k )}{ f_{ Y_ k} ( y_ k )}$$ I got: $$f_{ Y_ k | Y_ 1 ,..., Y_{ k−1}} ( y_ k | y_ 1 ,..., y_{ k−1} )=(k−1)!(λ y_ k)^{ 1−k}$$

and for c) using $$f_{ Y_ 1 ,.., Y_ k | Y_{ k+1} ,..., Y_ n} ( y_ 1 ,..., y_ k | y_{ k+1} ,..., y_ n )=\frac{ f_{ Y_ 1 ,.., Y_ n} ( y_ 1 ,..., y_ n )}{ f_{ Y_ 1 ,..., Y_ k} ( y_ 1 ,..., y_{ k −1})}$$

I got $$\frac{1}{ exp(λ( y_ n − y_ k ))}$$

Now I am stuck on how to show that it is uniformly distributed and to find the subset because I don't think that my answer looks like a uniformly distributed PDF...

• The subset when $Y_{k+1}=y_{k+1}$ is all points $(z_1,z_2,\ldots,z_k) \in \mathbb R^n$ where $0 \le z_1\le z_2 \le \ldots \le z_k \le y_{k+1}$ – Henry May 24 at 11:00
• Cross-posted at math.stackexchange.com/q/3683117/321264. – StubbornAtom May 24 at 15:18