grandes-ecoles 2016 QIII.D.3

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

For $\lambda \neq 1$, we denote $C_{n} = \sum_{k=0}^{n} \frac{(n\lambda)^{k}}{k!}$ and $D_{n} = \sum_{k=n+1}^{+\infty} \frac{(n\lambda)^{k}}{k!}$.
Determine $\lim_{n \rightarrow +\infty} \mathrm{e}^{-n\lambda} C_{n}$ if $\lambda < 1$ and $\lim_{n \rightarrow +\infty} \mathrm{e}^{-n\lambda} D_{n}$ if $\lambda > 1$.

For $\lambda \neq 1$, we denote $C_{n} = \sum_{k=0}^{n} \frac{(n\lambda)^{k}}{k!}$ and $D_{n} = \sum_{k=n+1}^{+\infty} \frac{(n\lambda)^{k}}{k!}$.

Determine $\lim_{n \rightarrow +\infty} \mathrm{e}^{-n\lambda} C_{n}$ if $\lambda < 1$ and $\lim_{n \rightarrow +\infty} \mathrm{e}^{-n\lambda} D_{n}$ if $\lambda > 1$.

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