Suppose $T_1, T_2, . . . , T_n$ is a sequence of independent, identically distributed random variables with the exponential distribution of the density function
$$p(x)=\begin{cases}e^{-x},~~x\ge0;\\0,~~~~x<0.\end{cases}$$
Let $S_n = T_1 + T_2 + · · · + T_n$. Find the distribution of the random vector
$$V_n=\bigg\{\frac{T_1}{S_n},\frac{T_2}{S_n},\cdots,\frac{T_n}{S_n}\bigg\}$$
We know that $T_1$ and $S_n-T_1$ are independent, $S_n-T_1$ follows the Gamma distribution $\Gamma(n-1,1)$, so we can calculate distribution of $\frac{T_1}{S_n}$. Since all elements are iid, we know every elements' distributions. However, the sum of all elements is $1$, so they are not independent. My question is, how to present the distribution of $V_n$, given that the sum of all elements in $V_n$ is $1$?
Best Answer
Let $U_1,\ldots ,U_{n-1}$ be $n-1$ independent variables, uniform in $[0,1]$. Arrange them in increasing order to obtain the order statistics $$U_{(1)}<U_{(2)}< \ldots <U_{(n-1)} \,.$$ Then the vector of gaps $$(U_{(1)}, U_{(2)}-U_{(1)}, \ldots, U_{(n-1)}-U_{(n-2)},1-U_{(n-1)}) \, $$ has the same (joint) distribution as the vector $$V_n=\bigg\{\frac{T_1}{S_n},\frac{T_2}{S_n},\cdots,\frac{T_n}{S_n}\bigg\} \,.$$
See Theorem 6.6 (c) in https://www.stat.purdue.edu/~dasgupta/orderstats.pdf