# distribution of fractions of partial sums of exponential random variables

Let $$X_1, X_2, ...$$ are iid exponential random variables, $$S_k=\sum_{i=1}^{k}X_i$$.

I want to find the distributions of $$S_k/S_n$$ for $$k=1,...,n-1$$.

i first used transformation $$Y_i=S_k/S_n , (k=1,...,n-1)$$ and $$Y_n=X_1+...+X_n$$ So that i assume Jacobian is $$y_n^{n-1}$$ since $$x_1=y_1y_n, x_2=y_2y_n-y_1y_n,...,x_{n-1}=y_{n-1}y_n-y_{n-2}y_n,x_n=y_n(1-y_{n-1}).$$

so $$f_{Y_1,...,Y_n}(y_1,...,y_n)=\lambda^ne^{-\lambda y_n}y_n^{n-1},\space y_1\in(0,1),y_2\in(y_1,1),...,y_n\in(0,\infty)$$

Is this right way to calculate the distribution of $$S_k/S_n=Y_k$$? i found tricky calculating these.. am i missing of mistaking something?

Each $$S_k$$ as the sum of iid. exponential $$X_i \sim \mathrm{Exp}(\lambda)$$ is $$\mathrm{Gamma}(k,\lambda)$$, regardless of you're using rate parameter of scale parameter.
The fraction is $$\displaystyle \frac{S_k}{S_n} = \frac{S_k }{S_k + S'_k}$$, with $$\displaystyle S'_k \equiv \sum_{i = k+1}^n X_i \sim \mathrm{Gamma}(n-k,\lambda)$$ that is also Gamma, and we have independence $$S_k \perp S'_k$$ since $$X_i$$ are iid.
Therefore $$\frac{S_k}{S_n}$$ as a fraction of Gamma over "same Gamma plus another Gamma" follows $$\mathrm{Beta}(k,n-k)$$.
If you really want to calculate the joint density (and then "integral out the rest" to get the marginals), you should start with $$n = 2$$ then do $$n = 3$$ to see the patterns. Don't dive into general $$n$$ directly unless you're already fairly with the relevant integrals.
• (+1) It is enough to derive the density of $\frac{S_k}{S_k+S_k'}$ using a change of variables since we know the distribution of both $S_k$ and $S_k'$ and they are independent. – StubbornAtom Oct 19 '18 at 16:48