grandes-ecoles 2017 QIII.A.5

grandes-ecoles · France · centrale-maths2__official Probability Generating Functions Radius of convergence and analytic properties of PGF

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$, with auxiliary power series $H_{X}(t) = \ln(\mathbb{P}(X = 0)) + \sum_{k=1}^{\infty} \lambda_{k} t^{k}$ defined for $t \in ]-\rho(X), \rho(X)[$.
For $t \in ]-\rho(X), \rho(X)[$, show $G_{X}^{\prime}(t) = H_{X}^{\prime}(t) G_{X}(t)$, then $G_{X}(t) = \exp\left(H_{X}(t)\right)$.

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$, with auxiliary power series $H_{X}(t) = \ln(\mathbb{P}(X = 0)) + \sum_{k=1}^{\infty} \lambda_{k} t^{k}$ defined for $t \in ]-\rho(X), \rho(X)[$.

For $t \in ]-\rho(X), \rho(X)[$, show $G_{X}^{\prime}(t) = H_{X}^{\prime}(t) G_{X}(t)$, then $G_{X}(t) = \exp\left(H_{X}(t)\right)$.

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