Posterior predictive distribution
If \(X|\lambda \sim \mathcal{E}(\lambda)\) with \(\lambda \sim \mathcal{G}(\alpha, \beta)\), then the posterior predictive probability density function, the expected value and variance of \(X\) are
\[f(x; \alpha, \beta) = \frac{\alpha \beta^{\alpha}}{(\beta + x)^{\alpha + 1}}, \quad x \ge 0.\]
\[\mathrm{E}[X] = \frac{\beta}{\alpha - 1}, \quad \mathrm{Var}[X] = \frac{\alpha \beta^2}{(\alpha - 1)^2 (\alpha - 2)},\]
where \(\mathrm{E}[X]\) is defined for \(\alpha > 1\) and
\(\mathrm{Var}[X]\) is defined for \(\alpha > 2\).
Proofs
Posterior predictive probability density function
\[\begin{split}f(x; \alpha, \beta) &= \int_0^{\infty} \lambda e^{-\lambda x} \frac{\beta^{\alpha} \lambda^{\alpha - 1} e^{-\beta \lambda}}{\Gamma(\alpha)} \mathop{d\lambda} = \frac{\beta^{\alpha}}{\Gamma(a)} \int_0^{\infty} \lambda^{\alpha} e^{-\lambda(\beta + x)} \mathop{d\lambda}\\
&= \frac{\beta^{\alpha}}{\Gamma(a)}\frac{\Gamma(a + 1)}{(\beta + x)^{\alpha + 1}} = \frac{\alpha \beta^{\alpha}}{(\beta + x)^{\alpha + 1}}.\end{split}\]
Note that this is the probability density function of the Lomax distribution, thus
\[X \sim \mathcal{Lomax}(\alpha, \beta),\]
see https://en.wikipedia.org/wiki/Lomax_distribution.
Posterior predictive expected value
\[\mathrm{E}[X] = \mathrm{E}[\mathrm{E}[X | \lambda]] = \mathrm{E}\left[\frac{1}{\lambda}\right],\]
The reciprocal of the gamma distribution follows an inverse gamma distribution with expected value \(\frac{\beta}{\alpha - 1}\).
Posterior predictive variance
Instead of directly applying well-known properties of the Lomax distribution, we use the law of total variance,
\[\begin{split}\mathrm{Var}[X] &= \mathrm{E}[\mathrm{Var}[X | \lambda]] + \mathrm{Var}[\mathrm{E}[X | \lambda]] = E\left[\frac{1}{\lambda^2}\right] + \mathrm{Var}\left[\frac{1}{\lambda}\right]\\
&= \frac{\beta^2}{(\alpha - 1)(\alpha - 2)} + \frac{\beta^2}{(\alpha - 1)^2(\alpha - 2)} = \frac{\alpha \beta^2}{(\alpha - 1)^2 (\alpha - 2)},\end{split}\]
where we use the second moment and the variance of the inverse gamma distribution.