grandes-ecoles 2018 Q29

grandes-ecoles · France · centrale-maths1__official Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof

We denote $p_{+} = \mathbb{P}(X' \in C_{+1})$ and $p_{-} = \mathbb{P}(X' \in C_{-1})$, with $p_{+} \geqslant p_{-}$. We set $\lambda = 1 - \frac{p_{-}}{p_{+}}$. Show that
$$\mathbb{E}\left(\exp\left(\frac{1}{8}d(X, C)^{2}\right)\right) \leqslant \frac{1}{2p_{+}}\left(1 + \exp\left(\frac{\lambda^{2}}{2}\right)(1 - \lambda)^{\lambda - 1}\right)$$

We denote $p_{+} = \mathbb{P}(X' \in C_{+1})$ and $p_{-} = \mathbb{P}(X' \in C_{-1})$, with $p_{+} \geqslant p_{-}$. We set $\lambda = 1 - \frac{p_{-}}{p_{+}}$. Show that

$$\mathbb{E}\left(\exp\left(\frac{1}{8}d(X, C)^{2}\right)\right) \leqslant \frac{1}{2p_{+}}\left(1 + \exp\left(\frac{\lambda^{2}}{2}\right)(1 - \lambda)^{\lambda - 1}\right)$$

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