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Let $X \sim \Gamma(m, p)$ with a shape parameter $m$ and a scale parameter $p$ and $Y \sim \Gamma(m, q)$ with a shape parameter $m$ and a scale parameter $q$, and let $X$ and $Y$ be independent.

What will be the PDF and CDF of $X+Y$? How can I solve it in MatLab?

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  • $\begingroup$ Notation comment: $\Gamma$ is used rather for denoting the gamma function and gamma distribution is rather written simply as "gamma". $\endgroup$
    – Tim ♦
    Jan 14, 2016 at 9:02

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The PDF of the sum is the convolution of the probability density functions. You can get the CDF by integrating that PDF. For more information see

https://en.wikipedia.org/wiki/Convolution_of_probability_distributions

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    $\begingroup$ This answer merely parrots the official line about convolutions without bothering to consider any of the difficulties that are encountered. $\endgroup$
    – Dilip Sarwate
    Jan 14, 2016 at 7:56
  • $\begingroup$ I second @DilipSarwate and think the answer deserves to be expanded. $\endgroup$
    – Richard Hardy
    Jan 14, 2016 at 8:25
  • $\begingroup$ Hi @DilipSarwate. With all due respect, that comment is at least as useless as my answer, because it does not expand on what those difficulties are. As an added bonus, your comment is mean-spirited. Submitting your own answer may be a better use of your time (and more helpful to the person who asked the question) than censuring me and leaving without further comment. $\endgroup$
    – linksys
    Jan 19, 2016 at 1:55
  • $\begingroup$ linksys, I have answered several questions about Gamma random variables on stats.SE including one about the sum of independent Gamma random variables with same scale parameter. When the scale parameters are different as in this particular question, there is more difficulty in getting at the final result (which is well-documented in the answers to the question"General sum of Gamma random variables", and so needs no further amplification from me). I am down voting your answer pending revision. The official reason for a down vote is "This answer is not useful", a description that I agree with. $\endgroup$
    – Dilip Sarwate
    Jan 19, 2016 at 2:18
  • $\begingroup$ @DilipSarwate, I appreciate this comment that moves toward something constructive. You may consider linking to the relevant content or informing the person who asked the question about its existence. I don't understand that last bit about down voting. Is there somewhere I can view the reason? $\endgroup$
    – linksys
    Jan 19, 2016 at 2:33

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