NB2

Solution 6.9 de l’exercice 3.3

\[\begin{eqnarray*} \Pr[S=s] &=& \int \Pr[S=s|\theta] f(\theta) d\theta \\ &=& \int_0^{\infty} \frac{(\lambda \theta)^{s_t} e^{-\lambda \theta}}{s!} \frac{\alpha^{\alpha}}{\Gamma(\alpha)} \theta^{\alpha - 1} e^{-\alpha \theta} d\theta \\ &=& \frac{\alpha^{\alpha} \lambda^{s}}{\Gamma(\alpha) s!} \int_0^{\infty} \theta^{s + \alpha - 1} e^{-\theta( \lambda + \alpha)} d\theta \\ &=& \frac{\alpha^{\alpha} \lambda^{s}}{\Gamma(\alpha) s!} \frac{\Gamma(s + \alpha)}{(\lambda + \alpha)^{s + \alpha}} \\ &=& \binom{\alpha + s - 1}{s} \left(\frac{\alpha}{\lambda + \alpha}\right)^{\alpha} \left(\frac{\lambda}{\lambda + \alpha}\right)^{s} \end{eqnarray*}\]