2. Let a particle on the $x$-axis initially be located at the origin. Assume that it is subjected to a random force every unit of time that causes its position to change, and it moves one unit to the right or left on the $x$-axis with probabilities $p$ and $q=1-\dot{p}$, respectively, but can only land on integer points on the $x$-axis. Find the probability that the particle is at integer point $k$ at time $t=n$.
Prompt Example 7.
$P\left(\xi_{n}=k\right)=\mathrm{C}_{n}^{r} p^{n-r} q^{r}$, where, $r=\frac{n-k}{2}$.