I have this distribution for $\theta$ $$\gamma \frac{1}{\theta^{n+2}} \exp\left\{-\frac{1}{\theta} \left(\frac12\sum_{i=1}^nX_i^2+a\right)\right\},$$ where $\gamma$ is a normalization factor independent of $\theta$, which looks quite similar to a Gamma dsitribution with parameters $n+3$ and $\frac12\sum_{i=1}^nX_i^2+a$, however the $\theta$ is inverted, which makes me unsure, since a "true" gamma distribution is given by
$$ \gamma \theta^{\alpha-1}\exp\left\{-\theta \beta \right\}.$$ What do you guys think?