Esp. et Variance NB2

Solution 6.10 de l’exercice 3.4

\[\begin{eqnarray*} E[S] &=& E[E[S|\Theta]] = E[\lambda \Theta] = \lambda E[\Theta] = \lambda \tfrac{\alpha}{\alpha}= \lambda \\ Var[S] &=& E[Var[S|\Theta]] + Var[E[S|\Theta]] \\ &=& E[\lambda \Theta] + Var[\lambda \Theta] \\ &=& \lambda E[\Theta] + \lambda^2 Var[\Theta] \\ &=& \lambda \tfrac{\alpha}{\alpha} + \lambda^2 \tfrac{\alpha}{\alpha^2} \\ &=& \lambda + \tfrac{\lambda^2}{\alpha} \end{eqnarray*}\]