Link between the Beta and Exponential distribution Let $n \geq 1$ be an integer. Let $X \sim \operatorname{Beta}(i, n - i + 1)$ where $i \in \{1, ..., n\}$. Therefore:
$$
X = \frac{A_n}{A_n + B_n}
$$
where
$$
A_n = \sum_{r = 1}^i Z_r, \qquad B_n = \sum_{r = i + 1}^{n + 1} Z_r
$$
where $Z_r$ are i.i.d. $\operatorname{Exp}(1)$ random variables.
Why?
PS.
This is from [^Arnold2008] in the proof of the theorem 8.5.1 page 223.
[^Arnold2008]: Arnold, B. C., Balakrishnan, N., & Nagaraja, H. N. (2008). A first course in order statistics. Society for Industrial and Applied Mathematics.

One way to see the connection between the two is in how both expressions connect to the uniform distribution.
See the following image where $z_i \sim exp(1)$ and $s_k = \sum_{i=1}^k z_i$.


*

*The exponential distribution is the waiting time or distance between events in a Poisson process. This makes the values of $s_k/s_{n+1}$ similar to $n$ uniform distributed variables.

*The beta distribution is an order distribution of the uniform distribution. This makes the position of a particular $s_k/s_{n+1}$ similar to a beta distribution.

