grandes-ecoles 2017 QI.A.4

grandes-ecoles · France · centrale-maths2__mp Probability Generating Functions Infinite divisibility and decomposability via PGF

Let $A(T) \in \mathbb{R}[T]$ be the polynomial: $A(T) = T^{4} + 2T + 1$.
a) Let $U(T)$ and $V(T)$ be two polynomials with non-negative real coefficients such that $U(T)V(T) = A(T)$. Show that one of the polynomials $U(T)$ or $V(T)$ is constant.
One may distinguish cases according to the values of the degrees of $U(T)$ and $V(T)$.
b) Deduce that there exists a decomposable random variable $X$ such that $X^{2}$ is not decomposable.
One may consider the polynomial $\frac{1}{4}A(T)$.

Let $A(T) \in \mathbb{R}[T]$ be the polynomial: $A(T) = T^{4} + 2T + 1$.

a) Let $U(T)$ and $V(T)$ be two polynomials with non-negative real coefficients such that $U(T)V(T) = A(T)$. Show that one of the polynomials $U(T)$ or $V(T)$ is constant.

One may distinguish cases according to the values of the degrees of $U(T)$ and $V(T)$.

b) Deduce that there exists a decomposable random variable $X$ such that $X^{2}$ is not decomposable.

One may consider the polynomial $\frac{1}{4}A(T)$.

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