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I would like a formula which allows me to input some value for a Gamma distribution random variable, and get back the total probability density up to that point.

In essence, I would like say, a value of 0.05, or 0.95 returned from the Gamma CDF.

How do I know what value for the random variable would give this? It appears non-trivial due to the Gamma function itself. Is there actually a solution?

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It's not analytically available. You can use numerical approaches/software. What you want to find is actually inverse CDF of quantile function in other words.

In Matlab you can use gaminv, in R, you can use qgamma, or in python you can use ppf in scipy.stats.

It's not easy to find a table but here is one with unit scale, with varying shape parameters (upto some limit of course). Unfortunately, it's not as easy as for standard normal, i.e. Z table. You can use scaling property of Gamma to convert Gamma's with different scales.

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    $\begingroup$ There are two definitions of standard Gammas and one is extensively tabulated: it's called the chi-squared distribution. $\endgroup$ – whuber Feb 19 at 14:13

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