quick question about the form of a posterior distribution.
Suppose that $\theta \sim Gamma(a, b)$ and that, given $\theta$, $Y$ has CDF $$F(Y\mid\theta) = 1 - e^{-\theta(e^y - 1)},\quad \theta>0.$$
If I didn't make any mistakes differentiating then, given $\theta$, $Y$ has PDF
$$\theta e^{y - \theta(e^y - 1)}.$$
So the posterior distribution of $\theta\mid Y$ would be
\begin{align*} p(\theta \mid Y) &\propto p(Y|\theta) p(\theta) \\ &=\theta e^{y -\theta(e^y - 1)} \cdot \frac{1}{\Gamma(a)}b^a\theta^{a- 1} e^{-b\theta} \\ &\propto \theta^a e^{ - \theta(b+ e^y - 1)} \end{align*}
which seems like a $Gamma(a+1, \, b + e^y -1)$ distribution.
Does this seem correct?
I think it seems strange to me since the likelihood
$$\theta e^{y - \theta(e^{y} - 1)}$$
is not one I've dealt with before.