grandes-ecoles 2020 Q2

grandes-ecoles · France · x-ens-maths-b__mp_cpge Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables

Let $n \geqslant 1$ be a natural integer, and let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$, $$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$ We define $$S_n = \frac{1}{n} \sum_{k=1}^{n} X_k$$ Show that $P[S_n \geqslant 0] \geqslant \frac{1}{2}$.

Let $n \geqslant 1$ be a natural integer, and let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$,
$$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$
We define
$$S_n = \frac{1}{n} \sum_{k=1}^{n} X_k$$
Show that $P[S_n \geqslant 0] \geqslant \frac{1}{2}$.

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