grandes-ecoles 2017 QI.B.2

grandes-ecoles · France · centrale-maths1__psi Central limit theorem

Suppose that $X$ admits a second moment. If $u$ and $v$ are two real numbers such that $u < E(X) < v$, determine the limit of the sequence $\left(\pi_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad \pi_{n} = P\left(nu \leqslant S_{n} \leqslant nv\right)$$

Suppose that $X$ admits a second moment. If $u$ and $v$ are two real numbers such that $u < E(X) < v$, determine the limit of the sequence $\left(\pi_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by
$$\forall n \in \mathbb{N}^{*}, \quad \pi_{n} = P\left(nu \leqslant S_{n} \leqslant nv\right)$$

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QI.C.1 QI.C.2 QI.C.3 QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QII.C.1 QII.C.2 QII.C.3