Equivalent ways of parametrizing Gamma distribution

gamma distributionscipy

From scipy documentation at https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gamma.html, Gamma distribution is written as

$$f(x, \alpha) = \frac{x^{\alpha – 1} e^{-x}}{\Gamma(\alpha)}$$

The doc also says this is equivalent to the more common way of parametrizing Gamma distribution

$$f(x, \alpha, \beta) = \frac{\beta^\alpha x^{\alpha – 1} e^{-\beta x}}{\Gamma(\alpha)}$$

but with a scale of $\frac{1}{\beta}$.

Can anyone provide more details to show how the two are equivalent?

I'm not sure if this is the right way to do it, but if I substitute $x = \beta y$ into the first equation, it seems there would be a factor $\beta$ missing compared to the 2nd equation.

Best Answer

This distribution $f(x, \alpha) = \frac{x^{\alpha - 1} e^{-x}}{\Gamma(\alpha)}$ is the distribution with a fixed scale parameter $1/\beta = \theta = 1$.

The article states further on

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, gamma.pdf(x, a, loc, scale) is identically equivalent to gamma.pdf(y, a) / scale with y = (x - loc) / scale

So, in the end, they put the second parameter back by the use of the scale parameter.

if I substitute $x = \beta y$ into the first equation, it seems there would be a factor $\beta$ missing compared to the 2nd equation.

If you transform the variable $x = \beta y$ you are sort of squeezing or stretching the density function. When you do this then you need to correct the height as well in order that the pdf integrates to a total area of 1.

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