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I have a set of exponentially distributed random variables $X_i \sim \exp(\mu_i)$ with rates being also random with some distribution. Is there a way to find the distribution (or the CDF if it is easier) of $Y$ the sum of $X_i$,

$$Y=\sum\limits_{i=1}^N X_i,$$

other than the following:

$$f_Y(x) = \sum\limits_{i=1}^N \prod\limits_{j=i} \frac{\mu_i}{\mu_j-\mu_i} \mu_i e^{-\mu_i x}. $$

I found the above expression earlier, where $\mu_i$ is different for each $X_i$, but in my case, the rates are random with some distribution $f_\mu (\mu)$.

Is there any other expression for the sum?

Edit: My Question was marked as duplicate, but I never found where the duplicate original was. As far as I can see, the closest questions are either dealing with identical rates of the exponential, which is not my case or had the formula that I stated, which is not helpful for me.

Xander Henderson
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A. Shokair
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  • you can get your answer here https://math.stackexchange.com/questions/655302/gamma-distribution-out-of-sum-of-exponential-random-variables – DRPR May 04 '18 at 22:18
  • I agree that it's a mistake to mark this as a duplicate. Even if there is an older question somewhere in the site with the same content, it's not the one that is linked. Here with different rates the problem is considerably more difficult. – Lee David Chung Lin May 18 '18 at 05:38
  • Currently I don't have enough reps to vote for anything. I posted on meta to make a dispute, then I was guided to post here. I hence deleted my "dispute-post" on meta. Let's see what will happen. – Lee David Chung Lin May 18 '18 at 06:36
  • Glad to see that it is reopened. I'm keeping the comments just for the record. – Lee David Chung Lin Jun 12 '18 at 13:12

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