Distribution of $Y = 1/X$ when $X\sim \text{Gamma}(\alpha, \beta). $ Suppose $X\sim \text{Gamma}(\alpha, \beta). $ Determine the distribution of $Y = 1/X.$
This is what I reached as a solution: the distribution of $Y = 1/X$ is a gamma distribution with shape parameter $\alpha+1$ and scale parameter $1/(\beta\cdot y).$

With
$$
Y = 1/X \overset{X >0}{\iff} X = 1/Y, \\
\frac{\mathrm d \, 1/y}{\mathrm d \, y} = -\frac{1}{y^2},
$$
and using the shape-rate parameterization, a change of variables yields
$$
\begin{align}
\mathop{f_Y}\left(y\right) 
&= \mathop{\text{Gamma}}\left(\frac 1 y;\alpha, \beta\right) \cdot \left| \frac{\mathrm d \, 1/y}{\mathrm d \, y} \right| \\
&= 
\frac{\beta^\alpha}{\mathop{\Gamma}\left(\alpha\right)} \left(\frac 1 y\right)^{\alpha -1} \exp\left(-\beta \cdot \frac 1 y\right) \cdot \left|-\frac{1}{y^2}\right| \\
&= 
\frac{\beta^\alpha}{\mathop{\Gamma}\left(\alpha\right)} y^{-\alpha - 1} \exp\left(-\frac \beta y\right) \;\; \forall \, y \in \mathbb{R}_{>0}
\end{align}
$$
for the probability density function $\mathop{f_Y}$ of $Y$.
What distribution does $\mathop{f_Y}$ correspond to?
