I am looking to solve the following problem : let assume that $(X_k)_{1\le k \le K}$ and $(Y_k)_{1\le k\le K}$ are independent exponential random variables with parameter 1. What is the cumulative distribution of function of $$Z= \sum_{k=1}^K \frac {X_k}{Y_k}$$

The case of $K=1$ the cumulative function could be found easily using the integration formula. However I can not generalize the result for any $K$. Can any one help me to find an idea ?

  • $\begingroup$ In principle, one could compute the density of the sum by convolution, using induction to find a closed form for an arbitrary positive integer $K$. $\endgroup$ – Math1000 Jun 22 '18 at 5:10
  • $\begingroup$ That what I am looking for a closed form. I can not intuitive one and prove it by induction. $\endgroup$ – Youem Jun 22 '18 at 5:12

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