In this part, $d$ equals 1 and we simply denote $0_d = 0$. Moreover, $p$ is an element of $]0,1[$, $q = 1 - p$ and the distribution of $X$ is given by $$P ( X = 1 ) = p \quad \text{and} \quad P ( X = - 1 ) = q .$$ We consider the functions $F$ and $G$ defined by the formulas $$\begin{aligned}
& \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\
& \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n }
\end{aligned}$$ For $x \in ]-1,1[$, give a simple expression for $G ( x )$. Express $P ( R = + \infty )$ as a function of $| p - q |$. Determine the distribution of $R$.
In this part, $d$ equals 1 and we simply denote $0_d = 0$. Moreover, $p$ is an element of $]0,1[$, $q = 1 - p$ and the distribution of $X$ is given by
$$P ( X = 1 ) = p \quad \text{and} \quad P ( X = - 1 ) = q .$$
We consider the functions $F$ and $G$ defined by the formulas
$$\begin{aligned}
& \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\
& \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n }
\end{aligned}$$
For $x \in ]-1,1[$, give a simple expression for $G ( x )$.
Express $P ( R = + \infty )$ as a function of $| p - q |$.
Determine the distribution of $R$.