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I tried to prove using the convulution approach but it didn't work

user324997
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    show us your workings so far. The site supports $\LaTeX$ – Robert Long Jun 08 '21 at 09:33
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    Please add the [tag:self-study] tag & read its [wiki](https://stats.stackexchange.com/tags/self-study/info). Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. – Stephan Kolassa Jun 08 '21 at 11:44
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    A partial answer at https://stats.stackexchange.com/questions/371768/distribution-of-sum-of-exponentials – kjetil b halvorsen Jun 08 '21 at 13:46
  • Because exponential variables are a special case of Gamma variables, your question is answered in the generalization at https://stats.stackexchange.com/questions/72479. – whuber Jun 08 '21 at 22:00

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Hint: Remember that if $X\sim \Gamma(\alpha,\beta)$ then X has a moment generating function given by $M_x(t)=(\frac{\beta}{\beta -t})^{\alpha}$ , $t<\beta$.

Asssume that $X_1,X_2,......X_n$ are independent and identically distributed exponential random variables. Let $Z=\sum_{i=1}^{n} X_i$, try to find $M_z(t)=E(e^{tZ})$

fuzzys3t
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