For $i \in \mathbb{N}^{*}$, let $Y _ { i }$ be the Bernoulli random variable indicating the event
$$Y _ { i } = \mathbf{1} \left( S _ { i } \notin \left\{ S _ { k } , 0 \leq k \leq i - 1 \right\} \right) .$$
Show that, for $i \in \mathbb{N}^{*}$:
$$P \left( Y _ { i } = 1 \right) = P ( R > i )$$
Deduce that, for $n \in \mathbb{N}^{*}$:
$$E \left( N _ { n } \right) = 1 + \sum _ { i = 1 } ^ { n } P ( R > i )$$