Let $\left( X _ { k } \right) _ { k \in \mathbf{N} ^ { * } }$ be independent random variables with the same distribution given by:
$$P \left( X _ { 1 } = - 1 \right) = P \left( X _ { 1 } = 1 \right) = \frac { 1 } { 2 }$$
For all $n \in \mathbf { N } ^ { * }$, we denote $S _ { n } = \sum _ { k = 1 } ^ { n } X _ { k }$.
Deduce that for all $n \in \mathbf { N } ^ { * }$:
$$E \left( \left| S _ { n } \right| \right) = \frac { 2 } { \pi } \int _ { 0 } ^ { + \infty } \frac { 1 - ( \cos ( t ) ) ^ { n } } { t ^ { 2 } } \mathrm {~d} t$$