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The following is from an article I'm reading and is the conditional density of a random variable that is distributed according to a gamma distribution, conditional on the value of a parameter $t$.

$$f(x\mid t)=\frac1{\Gamma(q)}\frac{q}{g(t)}\left\{\frac{qx}{g(t)}\right\}^{q-1}e^{-qx/g(t)}\qquad\text{for}\qquad 0\leq x<\infty$$

I am having trouble interpreting the expression. Specifically:

What do the curly braces mean? And, how do I translate the $q$ and the $g(t)$ into the $k$ and $\theta$ used as parameters on the Wikipedia page?

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From the best I can tell, curly brackets are used for expression grouping only. For formula gives the probability density function, for a fixed value of variable $t$. This density function is a $\Gamma$-distribution with shape parameter $k = q$ and scale parameter $\theta = \frac{g(t)}{q}$, in the convention adopted at the linked wikipedia page.

Because the mean of $\Gamma(k, \theta)$ is $\mu = k \theta$, and the variance is $k \theta^2$, it follows that $$ \mathbb{E}(X|t) = q \frac{g(t)}{q} = g(t) \qquad \mathbb{Var}(X|t) = q \left( \frac{g(t)}{q} \right)^2 = \frac{g(t)^2}{q} $$

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