I was doing a Bayesian exercise and found this exercise about prior, posterior, predictive density and so on. I successfully solved parts A and B without much difficulty, but I have no idea in how to proceed with C, I thought I could find $y_{n+2}|y_{n+1}$ and multiply it by what I found in B, but I failed. Any ideas?
a) If $y_1, \ldots, y_n$ is a random sample from $Y \sim \text{Exp}(k)$ with the probability density function $f(y|k) = ke^{-ky}$, $y$ > 0, and a priori $k \sim \text{Gamma}(m, b)$, where $m$ and $b$ are known, then the posterior of $k$ is given by: $$k|\mathbf{y} \sim \text{Gamma}(n + m, b + t)$$ where $t = \sum_{i=1}^{n} y_i$
b) Find the predictive density of $y_{n+1}|\mathbf{y}$:
The predictive density of $y_{n+1}$ given the data is: $$\frac{(n + m)(b + t)^{n + m}}{(y_{n+1} + b + t)^{n + m + 1}}$$
c) Find the joint predictive density of $y_{n+1}$ and $y_{n+2}$ given $\mathbf{y}$:
Answer:
\begin{equation} \frac{(n + m)(n + m + 1)(b + t)^{n + m}}{(y_{n+2} + y_{n+1} + b + t)^{n + m + 2}} \end{equation}