= Rate parameter of a gamma distribution
{title2=$r=1/\theta$}
In the shape-rate convention for a <gamma distribution>, the rate parameter $r>0$ appears in the exponential decay:
$$
f(x)=\frac{r^a}{\Gamma(a)}x^{a-1}e^{-rx},\qquad x>0,\quad a>0.
$$
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.
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