grandes-ecoles 2017 QII.C.1

grandes-ecoles · France · centrale-maths2__mp Probability Generating Functions Bounding probabilities or tail estimates via PGF

Let $X$ and $Y$ be two random variables defined on $(\Omega, \mathcal{A}, \mathbb{P})$ and taking values in $\mathbb{N}$.
a) Show that if $A$ and $B$ are events in $\mathcal{A}$, and if $\bar{A}$ and $\bar{B}$ are their respective complementary events, then $$|\mathbb{P}(A) - \mathbb{P}(B)| \leqslant \mathbb{P}(A \cap \bar{B}) + \mathbb{P}(\bar{A} \cap B)$$
b) Deduce that, for all $t \in [-1,1], \left|G_{X}(t) - G_{Y}(t)\right| \leqslant 2\mathbb{P}(X \neq Y)$.

Let $X$ and $Y$ be two random variables defined on $(\Omega, \mathcal{A}, \mathbb{P})$ and taking values in $\mathbb{N}$.

a) Show that if $A$ and $B$ are events in $\mathcal{A}$, and if $\bar{A}$ and $\bar{B}$ are their respective complementary events, then
$$|\mathbb{P}(A) - \mathbb{P}(B)| \leqslant \mathbb{P}(A \cap \bar{B}) + \mathbb{P}(\bar{A} \cap B)$$

b) Deduce that, for all $t \in [-1,1], \left|G_{X}(t) - G_{Y}(t)\right| \leqslant 2\mathbb{P}(X \neq Y)$.

Paper Questions

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