Rate parameter of a gamma distribution ($r=1/\theta$)

In the shape-rate convention for a gamma distribution, the rate parameter $r>0$ appears in the exponential decay:It is the reciprocal of the scale parameter $\theta =1/r$, not the shape $a$. If $X$ has this distribution, then $rX$ has shape $a$ and rate one. This follows by the change of variables formula for a probability density.

The mean is $a/r$ and the variance is $a/r^2$. Thus doubling the rate at fixed shape halves the mean and quarters the variance. When $x$ measures time, $r$ has units of inverse time. For positive integer shape $a$, $X$ can be the waiting time for the $a$th event of a rate-$r$ Poisson process, so the parameter is literally its event rate. Shape-rate and shape-scale software conventions must be distinguished before substituting numerical parameters.