# Posterior Predictive Distribution for Uniform Likelihood and Pareto Prior

I'm trying to find the posterior predictive distribution for data $$X_i, \dots X_n$$ from a a $$Uniform [0, \theta]$$ distribution. The prior distribution for $$\theta$$ is a $$Pareto[\alpha, \beta]$$ distribution:

I have found the posterior distribution to be: $$\pi(\theta \vert \boldsymbol{X}) \propto Pareto[\alpha+n, \max(X_i, \beta)]$$.

Now, when trying to find the posterior predictive distribution, I believe I have to find $$f(y|\boldsymbol{X}) = \int_{\Theta}{f(y|\theta)\pi(\theta|\boldsymbol{X})d\theta}$$

where y is the new data point.

I thought I might try the following:

$$f(y|\boldsymbol{X}) \propto \int_{\max(X_i, \beta)}^{\infty}\frac{1}{\theta}\frac{1}{\theta^{\alpha+n+1}}d\theta=\int_{\max(X_i, \beta)}^{\infty}\frac{1}{\theta^{\alpha+n+2}}d\theta$$

but I end up getting a weird answer, so I'm pretty sure this is incorrect. Could someone point me in the right direction? Thanks!