I need to proof that the Pareto random variable is a mixture of the Gamma and Exponential distribution but need help with one part of the proof.
Consider $X$ being Exponential with parameter $\lambda$ and $\Lambda$ being Gamma with parameters $\alpha$ and $\beta$. So we can say the mixture distribution of $X$ is
$$ \begin{align} f_{X|\alpha, \beta} &= \int_0^\infty \lambda e^{-\lambda x} \cdot \frac{1}{\Gamma (\alpha) \beta ^ \alpha} \lambda^{\alpha - 1} e^{-\frac \lambda \beta} d\lambda\\ &=\frac {1} {\Gamma (\alpha) \beta ^ \alpha} \int_0^\infty \lambda ^\alpha e^{-\lambda x - \lambda \frac 1 \beta} d\lambda\\ &=\frac{\Gamma (\alpha +1)}{\Gamma (\alpha) \beta^\alpha} \int_0^\infty \frac{\lambda ^\alpha e^{-\lambda x - \lambda \frac 1 \beta}}{e^{-\lambda}\lambda^\alpha} d\lambda\\&= \frac{\alpha}{\beta^\alpha}\int_0^\infty e^{\lambda \frac{\beta - x -1}{\beta}} d\lambda\\ &=\frac{\alpha}{\beta^\alpha}\begin{bmatrix} \frac{\beta}{\beta - x -1}e^{\lambda\frac{\beta - x - 1}{\beta}} d\lambda\end{bmatrix}^\infty _0\\&=?\\ &=\frac{\alpha}{\beta^\alpha}\begin{bmatrix}\frac{\beta}{\beta x+1}\end{bmatrix}^{\alpha +1} \end{align} $$
