It has been well known that if $X_1$,...,$X_n$ are independent exponential random variables with common parameter $\mu$, then we have a gamma distribution $\sum_{i = 1}^{n}X_i \sim \Gamma(n,\mu)$ with two parameters $(n, \mu)$.

My question is do we have $\sum_{i = 1}^{n}X_i \sim \Gamma(n,\sum_{j}\mu_j)$ if we have $X_i$'s are independent exponential random variables with parameter $\mu_i$?

As I explored the internet and found this post, which says that the answer to my above question is negative (also confirmed by using moment generating function method).

So I wonder if there is a name of the distribution for such sum of exponential random variables with different parameters?

  • $\begingroup$ Thanks, it's fixed. $\endgroup$ – Liäm Sep 22 '17 at 3:33

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