grandes-ecoles 2025 Q26

grandes-ecoles · France · mines-ponts-maths2__psi Discrete Probability Distributions Conditional Expectation and Total Expectation Applications
Show that
$$\mathbf { E } \left[ \Delta \tilde { S } _ { n } \right] = - \mathbf { E } \left[ \tilde { S } _ { n } p \left( \tilde { I } _ { n } \right) \right]$$
then deduce the equation satisfied by $\mathbf { E } \left[ \Delta \tilde { I } _ { n } \right]$.
Here $\Delta \tilde{S}_n = \tilde{S}_{n+1} - \tilde{S}_n$, $\Delta \tilde{I}_n = \tilde{I}_{n+1} - \tilde{I}_n$, $\Delta \tilde{R}_n = \tilde{R}_{n+1} - \tilde{R}_n$, and $\tilde{S}_n + \tilde{I}_n + \tilde{R}_n = M$ for all $n$. 
Show that

$$\mathbf { E } \left[ \Delta \tilde { S } _ { n } \right] = - \mathbf { E } \left[ \tilde { S } _ { n } p \left( \tilde { I } _ { n } \right) \right]$$

then deduce the equation satisfied by $\mathbf { E } \left[ \Delta \tilde { I } _ { n } \right]$.

Here $\Delta \tilde{S}_n = \tilde{S}_{n+1} - \tilde{S}_n$, $\Delta \tilde{I}_n = \tilde{I}_{n+1} - \tilde{I}_n$, $\Delta \tilde{R}_n = \tilde{R}_{n+1} - \tilde{R}_n$, and $\tilde{S}_n + \tilde{I}_n + \tilde{R}_n = M$ for all $n$.
Paper Questions
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