grandes-ecoles 2017 QII.A.2

grandes-ecoles · France · centrale-maths1__psi Discrete Random Variables Independence Proofs for Discrete Random Variables

Let $a$ be a real number. Let $m$ and $n$ be in $\mathbb{N}$.
a) Show that $S_{m+n} - S_{m}$ and $S_{n}$ have the same distribution.
b) Let $b$ be a real number. Show $P\left(S_{m+n} \geqslant (n+m)b\right) \geqslant P\left(S_{n} \geqslant nb\right) P\left(S_{m} \geqslant mb\right)$.

Let $a$ be a real number. Let $m$ and $n$ be in $\mathbb{N}$.

a) Show that $S_{m+n} - S_{m}$ and $S_{n}$ have the same distribution.

b) Let $b$ be a real number. Show $P\left(S_{m+n} \geqslant (n+m)b\right) \geqslant P\left(S_{n} \geqslant nb\right) P\left(S_{m} \geqslant mb\right)$.

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