Distribution of $ Y_i=\frac{X_i}{X_1+X_2+...+X_n}, i=1, 2, ..., n-1 $ where $X_1, X_2, ..., X_n$ are iid each with a standard exponential distribution

Question

Problem

Let $X_1 ;X_2;...;X_n$ denote independent distributed random variables, each with a standard exponential distribution. Let $$ Y_i=\frac{X_i}{X_1+X_2+...+X_n}, i=1, 2, ..., n-1 $$ Show that the distribution of $Y_1; Y_2;...; Y_{n-1}$ is uniform on the simplex in $R^{n-1}$, meaning that the pdf is $ \frac{1}{(n-1)!} $ in a suitable support.

My idea:

1. Since $Y_i$ is a function of $n$ independent distributed random variables I think that first, we need to find the joint distribution of $X_1, X_2, ..., X_n$ which is: $$f(X_1, X_2, ..., X_n)=\prod_{i=1}^{n}f(X_i)=\prod_{i=1}^{n}{\lambda}e^{-{\lambda}_{i}X_{i}}$$

2. Transform $(X_1, X_2, ..., X_n)\rightarrow(U_1, U_2, ..., U_n)$ and transform them to determine the Jacobian.

How should I define $U_i$ variables?

Please can you help me to solve the problem?