grandes-ecoles 2016 QIII.C.4

grandes-ecoles · France · centrale-maths1__pc Central limit theorem

We fix two real numbers $a$ and $b$ such that $a < b$. For all integers $n \geqslant 1$ such that $a + \sqrt{n\lambda} > 0$, we define $$I_{n} = \{k \in \mathbb{N} \mid n\lambda + a\sqrt{n\lambda} \leqslant k \leqslant n\lambda + b\sqrt{n\lambda}\}$$
Express, in the form of an integral, $\lim_{n \rightarrow +\infty} \sum_{k \in I_{n}} \frac{(n\lambda)^{k}}{k!} \mathrm{e}^{-n\lambda}$.

We fix two real numbers $a$ and $b$ such that $a < b$. For all integers $n \geqslant 1$ such that $a + \sqrt{n\lambda} > 0$, we define
$$I_{n} = \{k \in \mathbb{N} \mid n\lambda + a\sqrt{n\lambda} \leqslant k \leqslant n\lambda + b\sqrt{n\lambda}\}$$

Express, in the form of an integral, $\lim_{n \rightarrow +\infty} \sum_{k \in I_{n}} \frac{(n\lambda)^{k}}{k!} \mathrm{e}^{-n\lambda}$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.B.1 QI.B.2 QI.C.1 QI.C.2 QII.A QII.B.1 QII.B.2 QII.B.3 QII.C.1 QII.C.2 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3 QIII.C.4 QIII.C.5 QIII.C.6 QIII.D.1 QIII.D.2 QIII.D.3 QIII.E.1 QIII.E.2 QIII.F