grandes-ecoles 2018 Q38

grandes-ecoles · France · centrale-maths1__pc Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables and $S_{n} = \sum_{j=1}^{n} X_{j}$. For $x$ real, $\lfloor x \rfloor$ denotes the integer part of $x$. For all real numbers $\delta > 0$ and $\tau > 0$, calculate $\mathbb{V}\left(\delta S_{\lfloor 1/\tau \rfloor}\right)$, the variance of the random variable $\delta S_{\lfloor 1/\tau \rfloor}$.

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables and $S_{n} = \sum_{j=1}^{n} X_{j}$. For $x$ real, $\lfloor x \rfloor$ denotes the integer part of $x$. For all real numbers $\delta > 0$ and $\tau > 0$, calculate $\mathbb{V}\left(\delta S_{\lfloor 1/\tau \rfloor}\right)$, the variance of the random variable $\delta S_{\lfloor 1/\tau \rfloor}$.

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