grandes-ecoles 2020 Q13

grandes-ecoles · France · mines-ponts-maths2__mp_cpge Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process

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 $$P ( X = 1 ) = p \quad \text{and} \quad P ( X = - 1 ) = q .$$ For $n \in \mathbb{N}$, determine $P \left( S _ { 2 n + 1 } = 0 \right)$ and justify the equality: $$P \left( S _ { 2 n } = 0 \right) = \binom { 2 n } { n } ( p q ) ^ { n }$$

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
$$P ( X = 1 ) = p \quad \text{and} \quad P ( X = - 1 ) = q .$$
For $n \in \mathbb{N}$, determine $P \left( S _ { 2 n + 1 } = 0 \right)$ and justify the equality:
$$P \left( S _ { 2 n } = 0 \right) = \binom { 2 n } { n } ( p q ) ^ { n }$$

Paper Questions

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