Say I have this situation with an exponential distribution and it's gamma conjugates:
$y\mid\lambda \sim exp(\lambda)$
$\lambda \sim gamma(\theta,\beta)$
$\lambda \mid y,\theta,\beta \sim gamma(\theta + 1, \beta + y)$
A trial shows that $y>x$, (where $x$ is just a constant) and we'd like to update $\lambda$. Am I correct to think that the posterior density would be given by the following equation?
$p(\lambda\mid \theta,\beta,y>x)= \int_{y=x}^\infty 1- gamcdf(x|\theta+1,\beta+y)dy$
Is there are better way to do this?