Suppose $X_n$ is a ''lazy'' asymmetric random walk such that, at each step, $X_{n+1} = X_n + 1$ with probability $\alpha$, $X_{n+1} = X_n - 1$ with probability $\beta$, and $X_{n+1} = X_n$ with probability $1-\alpha-\beta$. Assume that $X_0 = 0$ and $\alpha + \beta < 1$.
What is the correct approach to computing, for a given $n$, quantities like $E(X_n | X_n \geq 0)~,$ $P(X_n \geq 0)~,$ or, at least $E[X_n 1(X_n \geq 0)]$.
I would also appreciate pointers to good resources that talk about properties of such random walks - almost everything I have found talks exclusively about the ''symmetric'' case where $\alpha+ \beta = \frac{1}{2}$.