Let $X\sim Exp(\lambda_1)$ and $Y\sim Exp(\lambda_2)$, I am trying to find the distribution of $Z = X+Y$. I understand that $f_z(z)=\frac{\lambda_1 \lambda_2}{\lambda_2-\lambda_1} \left(\exp[-\lambda_1 z] - \exp[-\lambda_2 z]\right)$, but I am struggling with mapping this back to a distribution.
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To specify a distribution, it suffices to find the PDF or CDF, so you are done. Not every distribution has a name, and I don't believe this one does. – angryavian Mar 15 '19 at 19:19
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But I thought that sum of 2 exponential distribution would create a gamma distribution – VincentN Mar 15 '19 at 19:32
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perhaps this link will help https://math.stackexchange.com/questions/635443/sum-of-exponential-random-variable-with-different-means – Matthew Liu Mar 15 '19 at 19:49
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@VincentN It is a gamma distribution when $\lambda_1 = \lambda_2$. – angryavian Mar 15 '19 at 20:06
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oh okay so I made a mistake there – VincentN Mar 15 '19 at 20:20