grandes-ecoles 2017 QIII.A.4

grandes-ecoles · France · centrale-maths2__mp 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 $\mathbb{P}(X = 0) > 0$.
Show that the power series $\sum \lambda_{k} t^{k}$ has a radius of convergence $\rho(X)$ greater than or equal to $\mathbb{P}(X = 0)$. For all real $t$ in $]-\rho(X), \rho(X)[$, we set $$H_{X}(t) = \ln(\mathbb{P}(X = 0)) + \sum_{k=1}^{\infty} \lambda_{k} t^{k}$$

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$.

Show that the power series $\sum \lambda_{k} t^{k}$ has a radius of convergence $\rho(X)$ greater than or equal to $\mathbb{P}(X = 0)$. For all real $t$ in $]-\rho(X), \rho(X)[$, we set
$$H_{X}(t) = \ln(\mathbb{P}(X = 0)) + \sum_{k=1}^{\infty} \lambda_{k} t^{k}$$

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