# Sum of Exponential and Gamma Distributions [duplicate]

I have been learning sums of distributions and understand that the sum of exponential distributions with parameter B is a gamma distribution with parameters a=1 and B.

However, I need to figure out: What is the sum of X, an exponential distribution with parameter 0.2, and Y, a gamma distribution with parameters 3 and 0.2. I THINK it would be a gamma distribution based off of the previous knowledge but cannot find anything on this or how to do this.

• As exponential is a special case of gamma you can find answer here: stats.stackexchange.com/questions/72479/… – kjetil b halvorsen Mar 15 at 14:23
• Does this answer your question? How do gamma distributions add and what would that model? – kjetil b halvorsen Mar 15 at 14:24
• I'm not sure. Honestly I don't know enough of the terminology to understand. I'm at very basic, just starting to learn this level. Would the sum just be a gamma distribution with parameters 3 and 0.2? – Mdf12 Mar 17 at 14:14
• The gamma distribution has two different parameterizations. One with a rate parameter, the other with a scale parameter. So, when you ask for 0.2 as a parameter, you must specify whether that is a rate, e.g., in per min, or a scale, e.g., in min. – Carl Mar 30 at 3:42

I will show how to get an answer here using results from the duplicate Q. First, note from wiki on gamma distribution that there are two commonly used parametrizations, I will assume the shape-scale parametrization (with $$k, \theta$$) as that seems most used, and is the one implicitly used in the duplicate question. So $$X\sim \mathcal{Exp}(0.2)=\mathcal{Gamma}(k=1,\theta=0.2)$$ so the distribution of the sum is $$\mathcal{Gamma}(1+3,0.2)$$ using the result from answer by @whuber.
And, since for $$Y$$ the shape parameter $$k=3$$ is an integer, $$Y$$ itself is (can be represented as ...) a sum of three independent exponential random variables, see Distribution of sum of exponentials