grandes-ecoles 2025 QI.2

grandes-ecoles · France · x-ens-maths-c__mp Discrete Probability Distributions Proof of Probabilistic Inequalities or Bounds

For all $N \geq 1$, give an example of random variables satisfying hypotheses $$\mathbb{P}(|X_n| \leq K) = 1, \quad \mathbb{E}[X_n] = 0, \quad \operatorname{Var}(X_n) \leq 1$$ and such that $\mathbb{P}(|S_N| \geq N) \geq 1/2$, where $S_N := X_1 + \cdots + X_N$.

For all $N \geq 1$, give an example of random variables satisfying hypotheses
$$\mathbb{P}(|X_n| \leq K) = 1, \quad \mathbb{E}[X_n] = 0, \quad \operatorname{Var}(X_n) \leq 1$$
and such that $\mathbb{P}(|S_N| \geq N) \geq 1/2$, 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