There are $k$ number of sorted random variables $X_1 > \cdots > X_k$ with magnitude and they follow exponential distribution $\mu e^{-\mu}$.
I can get the values of $X_1,\cdots,X_k$ with matlab experimentally.
My method is to generate $K$ number of random variables with 1000 iterations and sorting them. Then, I can find the average values of random variables as $\overline{X}_1 = \frac{1}{1000}\sum_{i=0}^{1000}X_1^i,\cdots, \overline{X}_k =\frac{1}{1000}\sum_{i=0}^{1000}X_k^i$, where $X_l^i$ is $l$th random variable with magnitude in $i$th iteration
However, I wonder if I can get the values of these random variables mathematically.
My question is to find patterns of random variables $\overline{X}_1, \cdots, \overline{X}_k$.
When $\mu$ and $K$ are known, is there a closed form of sorted random variables $\overline{X}_1, \cdots, \overline{X}_k$?
example) $\overline{X}_1 = \frac{K}{\mu}, X_2 =\frac{K-1}{\mu},\cdots, \overline{X}_k =\frac{1}{\mu} $