We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$, and $u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$. Show, for all $n \in \mathbb{N}^{*}$, $$E\left(\left|S_{n}\right|\right) = \frac{2}{\pi} u_{n}$$ We will use the integral expression for the absolute value obtained in question I.A.5.
We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$,
$$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$
For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$, and $u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$.
Show, for all $n \in \mathbb{N}^{*}$,
$$E\left(\left|S_{n}\right|\right) = \frac{2}{\pi} u_{n}$$
We will use the integral expression for the absolute value obtained in question I.A.5.