I have a biased $2D$ random walk $(X_n,Y_n)$ such that in the x-direction, it either moves one step to the right or left. $(X_{n+1} = X_n+1$ with probability $\alpha$ and $X_{n+1} = X_n-1$ with probability $1-\alpha$. $Y_n$ either stays where it is or moves one up. ($Y_{n+1} =Y_n$ with probability $\beta$ and $Y_{n-1} = Y_n+1$ with probability $1-\beta$.
I define a hitting time as $\tau = min(X_n = a, Y_n = b)$, essentially this walk hitting a box of height $b$ and length $a$ with its bottom left corner at the origin, and $(X_0,Y_0) = (0,0)$.
I want the expectation of this hitting time, and the only way I know how to do it is from the definition, $\mathbb{E}[\tau] = \sum_{j=1}^{\infty} j\mathbb{P}(\tau = j)$.
The vertical direction is easier, as $\mathbb{P}(Y_j = b) = \binom{j}{b}(\beta)^{j-b}(1-\beta)^{b}$ is binomially distributed.
I am however having trouble calculating $\mathbb{P}(X_j = a)$ probability of reaching $a$ for the first time after $j$ steps.
I am also not sure how to proceed in actually finding $\mathbb{P}(\tau = j)$, the distribution of the minimum of the $x$ and $y$ hitting times. Any hints would be appreciated or an alternative method.