grandes-ecoles 2015 QIII.A.2

grandes-ecoles · France · centrale-maths1__psi Probability Generating Functions Deriving moments or distribution from a PGF

Let $\mu$ be a probability distribution characterized by the sequence $(p_k)_{k\in\mathbb{N}}$ with $\sum_{k=0}^{+\infty}p_k=1$ and $p_0+p_1<1$. We define the Galton-Watson process with $Y_0=1$ and the recurrence above. We assume that every random variable following distribution $\mu$ has expectation equal to $m$ and a variance.
Express, for $n\in\mathbb{N}$, the expectation of $Y_n$ in terms of $m$ and $n$.

Let $\mu$ be a probability distribution characterized by the sequence $(p_k)_{k\in\mathbb{N}}$ with $\sum_{k=0}^{+\infty}p_k=1$ and $p_0+p_1<1$. We define the Galton-Watson process with $Y_0=1$ and the recurrence above. We assume that every random variable following distribution $\mu$ has expectation equal to $m$ and a variance.

Express, for $n\in\mathbb{N}$, the expectation of $Y_n$ in terms of $m$ and $n$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.B QI.C QI.D.1 QI.D.2 QI.E.1 QI.E.2 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.B QII.C.1 QII.C.2 QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.B.1 QIII.B.2 QIII.C.1 QIII.C.2 QIII.C.3 QIV.A QIV.B QIV.C QIV.D QIV.E QIV.F QIV.G QV.A QV.B.1 QV.B.2 QV.C.1 QV.C.2 QV.D.1 QV.D.2 QV.E QV.F