I am working on the use of distributed delay applied to pharmacometric models. Specifically, the delay kernel I am interested in is the Gamma distribution, with non-integer shape.

The historical values $y$ are a simple table of values at points in time.

Calculation of a distributed delay requires convolving the Gamma distribution with the $y(t)$ function.

I have done this by modeling $y(t)$ as a series of rectangles, and the convolution with each rectangle is very simple, using the CDF of Gamma.

However, it would be more accurate if I could treat $y(t)$ as a series of trapezoids. I could do this if I could find an expression for the convolution of the Gamma with a simple right triangle distribution, but my math skills are not up to the task.

Does anyone know how to do this, or can point me to a good reference?

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    $\begingroup$ I was able to get an expression for the convolution of a Gamma and a Triangular distribution using a CAS. The expressions are very unwieldy, though. Here is a document containing the formulas for the pdf and cdf, respectively. $\endgroup$ – COOLSerdash Mar 18 '17 at 21:09
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    $\begingroup$ Why don't you choose a more tractable model for the data? That will avoid having to do this (messy) calculation. $\endgroup$ – whuber Mar 18 '17 at 22:58
  • $\begingroup$ @whuber: I'm not sure if you mean the gamma distribution or $y(t)$. The gamma distribution is just what's needed to model absorption delay. The $y(t)$ is just a tabular sampling of past values of $y$. $\endgroup$ – Mike Dunlavey Mar 18 '17 at 23:49
  • $\begingroup$ @COOLSerdash: Thanks. What is a CAS? I need a magnifying glass to read the file, and besides my virus scanner objected to it :) $\endgroup$ – Mike Dunlavey Mar 18 '17 at 23:51
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    $\begingroup$ @Mike Computer Algebra System $\endgroup$ – Glen_b Mar 19 '17 at 9:16

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