Let $(X_k)$ be a sequence of random variables as following : $P(X_i=-1)=q$ and $P(X_i=1)=1-q$. I want to calculate $$E[\inf\{n:X_1+X_2+...+X_n>0\}]$$
My work so far
Let's denote $Y(x)=\inf\{X_1+X_2+...+X_n>x\}$
Then we want to calculate $E[Y(1)]$
$$E[Y(x)]=E[Y(x)|X_1=-1]P(X_1=-1)+E[Y(x)|X_1=1]P(X_1=1)=$$ $$=E[Y(x)|X_1=-1]q+1-q$$
And now I have a little troubles with calculating $E[Y(x)|X_1=-1]$. Could you please tell me if I'm going in right direction ?
Let $S_n:=k+\sum_{i=1}^n X_i$ and $T_z:=\inf\{n\ge 1: S_n=z\}$. Then $$ \mathsf{P}_0(T_1=1)=1-q, $$ and for $n\ge 1$, \begin{align}\label{1}\tag{1} \mathsf{P}_0(T_1=2n+1)&=q\mathsf{P}_{-1}(T_1=2n)=\frac{q}{n}\mathsf{P}_{-2}(S_{2n}=0) \\ &=\frac{q}{n}\binom{2n}{n-1}q^{n-1}(1-q)^{n+1}, \end{align} where $\mathsf{P}_k$ is the law of $S_n$ starting at $k$. Therefore, for $q<1/2$, \begin{align} \mathsf{E}T_1&=\sum_{n\ge 0}(2n+1)\mathsf{P}_0(T=2n+1) \\ &=(1-q)+q\sum_{n\ge 1}\frac{2n+1}{n}\binom{2n}{n-1}q^{n-1}(1-q)^{n+1} \\ &=\frac{1}{1-2q}. \end{align} Obviously, for $q\ge 1/2$, $\mathsf{E}T=\infty$.
For the second equality in \eqref{1} we used the fact that $$ \mathsf{P}_{-1}(T_1=k)=\mathsf{P}_{-2}(T_0=k)=\frac{2}{k}\mathsf{P}_{-2}(S_k=0). $$