# Sum of 2 exponential distribution with different parameters

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.

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