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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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  • $\begingroup$ 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. $\endgroup$ – angryavian Mar 15 at 19:19
  • $\begingroup$ But I thought that sum of 2 exponential distribution would create a gamma distribution $\endgroup$ – VincentN Mar 15 at 19:32
  • $\begingroup$ perhaps this link will help math.stackexchange.com/questions/635443/… $\endgroup$ – Matthew Liu Mar 15 at 19:49
  • $\begingroup$ @VincentN It is a gamma distribution when $\lambda_1 = \lambda_2$. $\endgroup$ – angryavian Mar 15 at 20:06
  • $\begingroup$ oh okay so I made a mistake there $\endgroup$ – VincentN Mar 15 at 20:20

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