grandes-ecoles 2025 QI.3

grandes-ecoles · France · x-ens-maths-c__mp Continuous Probability Distributions and Random Variables Expectation and Moment Inequality Proof

We assume that the random variables $X_1, \ldots, X_N$ are pairwise uncorrelated, that is: $$\forall 1 \leq m, n \leq N, \quad n \neq m \Rightarrow \mathbb{E}[X_n X_m] = 0.$$ Prove that $$\mathbb{E}\left[|S_N|^2\right] \leq N.$$ Deduce that, for all $t > 0$, $$\mathbb{P}\left(|S_N| > t\sqrt{N}\right) \leq \frac{1}{t^2}$$ where $S_N := X_1 + \cdots + X_N$.

We assume that the random variables $X_1, \ldots, X_N$ are pairwise uncorrelated, that is:
$$\forall 1 \leq m, n \leq N, \quad n \neq m \Rightarrow \mathbb{E}[X_n X_m] = 0.$$
Prove that
$$\mathbb{E}\left[|S_N|^2\right] \leq N.$$
Deduce that, for all $t > 0$,
$$\mathbb{P}\left(|S_N| > t\sqrt{N}\right) \leq \frac{1}{t^2}$$
where $S_N := X_1 + \cdots + X_N$.

Paper Questions

QI.1 QI.2 QI.3 QI.4 QI.5 QI.6 QII.1 QII.2 QII.3