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I understand that when α is an integer, the Gamma(α,β) distribution is the distribution of the length of time you have to wait until a total of α events have occurred, where the time between each event follows an Exponential(β) distribution.

However, does this property require each exponential distribution to be independent?

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Yes, of course, independence is required. For a simple counterxample, take the case where all the exponentials are actually the same variable.

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  • $\begingroup$ Thanks for clearing my doubts. :) $\endgroup$ – James May 8 '18 at 15:32

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