The Problem:
Consider $X_1, ..., X_5$, $5$ iid, exponentially distributed random variables with $\lambda = 1$. Let $Y_1 \lt Y_2 \lt Y_3 \lt Y_4 \lt Y_5$ be their order statistics.
Find the distribution of $ \sum_{i=1}^{4} Y_i $.
SOLUTION:
$$ \sum_{i=1}^{4} Y_i \sim \text{Expo}(5) + \text{Expo}(4) + \text{Expo}(3) + \text{Expo}(2). \qquad (*)$$
How does this work? I understand that $Y_j - Y_{j-1} \sim \text{Expo}((n-j+1)\lambda)$, and therefore, for each $Y_i$, we have that $Y_i = \sum_{j=1}^{i} (Y_j - Y_{j-1})$; but, that means that the desired sum should be a sum of $10$ exponentially distributed random variables (because each $Y_i$ is the sum of $i$ exponential RVs) instead of $4$. What am I missing here?