I have the feeling this should be doable, or at least have an approximation, but I'm failing to find one.

Let's consider a random variable $C$, that belongs to a Truncated Exponential distribution between 0 and 1. If we observe $n$ i.i.d. variables $C$, what is the distribution of the harmonic mean of this set? Formally, what is the PDF of

$P = \dfrac{n}{\sum_{i=1}^n \frac{1}{C_i}}$

I have the feeling that this question is getting close to that answer, but I'm not sure.

Now, the may reason why I started by this was to introduce the following complication. If to each random variable $C_i$ there's a weight $w$ associated and this weights come from another Truncated Exponential distribution, what is the Weighted harmonic mean?

$P2 = \dfrac{\sum_{i=1}^n w_i}{\sum_{i=1}^n \frac{wi}{C_i}}$

I'm getting that $P2$ and $P1$ are equivalents? Is this correct?


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