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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

$$ \begin{align} f_{X|\alpha, \beta} &= \int_0^\infty \lambda e^{-\lambda x} \cdot \frac{1}{\Gamma (\alpha) \beta ^ \alpha} \lambda^{\alpha - 1} e^{-\frac \lambda \beta} d\lambda\\ &=\frac {1} {\Gamma (\alpha) \beta ^ \alpha} \int_0^\infty \lambda ^\alpha e^{-\lambda x - \lambda \frac 1 \beta} d\lambda\\ &=\frac{\Gamma (\alpha +1)}{\Gamma (\alpha) \beta^\alpha} \int_0^\infty \frac{\lambda ^\alpha e^{-\lambda x - \lambda \frac 1 \beta}}{e^{-\lambda}\lambda^\alpha} d\lambda\\&= \frac{\alpha}{\beta^\alpha}\int_0^\infty e^{\lambda \frac{\beta - x -1}{\beta}} d\lambda\\ &=\frac{\alpha}{\beta^\alpha}\begin{bmatrix} \frac{\beta}{\beta - x -1}e^{\lambda\frac{\beta - x - 1}{\beta}} d\lambda\end{bmatrix}^\infty _0\\&=?\\ &=\frac{\alpha}{\beta^\alpha}\begin{bmatrix}\frac{\beta}{\beta x+1}\end{bmatrix}^{\alpha +1} \end{align} $$

bryan.blackbee
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  • I think the equality between second and third row is not correct. – Juho Kokkala Jan 31 '14 at 15:28
  • Hello @JuhoKokkala, any idea how to fix it? Well, I used the identity $\Gamma (\alpha +1) = \int_0^\infty e^{-\lambda}\lambda^\alpha dt$ – bryan.blackbee Jan 31 '14 at 15:30
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    But that identity does not imply that dividing the integrand by $e^{-\lambda}\lambda^\alpha$ and multiplying the result by $\Gamma(\alpha+1)$ cancel out - note that $\int (f(x) / g(x)) dx$ is not generally equal to $\int f(x) dx~/ \int g(x) dx$. – Juho Kokkala Jan 31 '14 at 15:57
  • could you propose an answer and help me with the solution and proof? – bryan.blackbee Jan 31 '14 at 16:07
  • Sure! I have solved it successfully. Thank you for your help! – bryan.blackbee Feb 02 '14 at 06:58
  • This is not a "mixture of exponential and gamma"; rather it is a gamma mixture of exponentials. The gamma distribution and the exponential distribution do not play symmetrical roles here. – Michael Hardy Aug 25 '18 at 20:48

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compound of gamma and exponential distribution

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