Sum of Exponential and Gamma Distributions

Question

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: https://stats.stackexchange.com/questions/72479/generic-sum-of-gamma-random-variables — kjetil b halvorsen, Mar 15 '20 at 14:23
Does this answer your question? How do gamma distributions add and what would that model? — kjetil b halvorsen, Mar 15 '20 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 '20 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 '20 at 03:42

kjetil b halvorsen · Accepted Answer · 2020-03-20T12:26:30.060

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

Sum of Exponential and Gamma Distributions

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