grandes-ecoles 2020 Q15

grandes-ecoles · France · mines-ponts-maths2__mp_cpge Discrete Probability Distributions Limit and Convergence of Probabilistic Quantities

In this part, $d$ equals 1 and we simply denote $0_d = 0$. Moreover, $p$ is an element of $]0,1[$, $q = 1 - p$ and the distribution of $X$ is given by $$P ( X = 1 ) = p \quad \text{and} \quad P ( X = - 1 ) = q .$$ We assume that $$p = q = \frac { 1 } { 2 }$$ Give a simple equivalent of $P ( R = 2 n )$ as $n$ tends to $+ \infty$. Deduce a simple equivalent of $E \left( N _ { n } \right)$ as $n$ tends to $+ \infty$.

In this part, $d$ equals 1 and we simply denote $0_d = 0$. Moreover, $p$ is an element of $]0,1[$, $q = 1 - p$ and the distribution of $X$ is given by
$$P ( X = 1 ) = p \quad \text{and} \quad P ( X = - 1 ) = q .$$
We assume that
$$p = q = \frac { 1 } { 2 }$$
Give a simple equivalent of $P ( R = 2 n )$ as $n$ tends to $+ \infty$. Deduce a simple equivalent of $E \left( N _ { n } \right)$ as $n$ tends to $+ \infty$.

Paper Questions

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