Consider an organism moving on a one-dimensional lattice of spacing $a$, taking steps either to the right or the left at regular time intervals $\tau$. In this random walk there is a slight bias to the right, that is the probabilities of moving to the right and left, $\alpha$ and $\beta$, are such that $\alpha-\beta=\epsilon$, where $0<\epsilon \ll 1$. Write down the appropriate master equation for this process. Show by taking the continuum limit in space and time that $p(x, t)$, the probability that an organism initially at $x=0$ is at $x$ after time $t$, obeys

$$\frac{\partial p}{\partial t}+V \frac{\partial p}{\partial x}=D \frac{\partial^{2} p}{\partial x^{2}}$$

Express the constants $V$ and $D$ in terms of $a, \tau, \alpha$ and $\beta$.