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If a bunch of random variables $X_i$ are independently and identically distributed with an exponential distribution, their sum apparently follows a Gamma distribution.

But doesn't the central limit theorem imply that (for $X_i$ of any distribution with mean zero and variance $\sigma^2$), the sum $\sum_{i=1}^n X_i$ will become approximately normally distributed $~N(0,n\sigma^2)$ for large enough $n$ ?

Obviously I am missing something basic, but what's going on? How can the sum of i.i.d. exponential random variables have a Gamma distribution, but also be converging to normality?

2 Answers 2