grandes-ecoles 2018 Q27

grandes-ecoles · France · centrale-maths1__mp Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof

We denote
$$p_{+} = \mathbb{P}(X' \in C_{+1}) \quad \text{and} \quad p_{-} = \mathbb{P}(X' \in C_{-1})$$
We will assume, without loss of generality, that $p_{+} \geqslant p_{-}$. Using the induction hypothesis, justify that
$$\mathbb{E}\left(\left.\exp\left(\frac{1}{8} d(X, C)^{2}\right)\right\rvert \, \varepsilon_{n} = 1\right) \leqslant \frac{1}{p_{+}}$$

We denote

$$p_{+} = \mathbb{P}(X' \in C_{+1}) \quad \text{and} \quad p_{-} = \mathbb{P}(X' \in C_{-1})$$

We will assume, without loss of generality, that $p_{+} \geqslant p_{-}$. Using the induction hypothesis, justify that

$$\mathbb{E}\left(\left.\exp\left(\frac{1}{8} d(X, C)^{2}\right)\right\rvert \, \varepsilon_{n} = 1\right) \leqslant \frac{1}{p_{+}}$$

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