Solution
= Solution
{c}
Only upward and downward moves change the value, so the <conditional expectation> is
$$
\mathbb E[R_{k+1}\mid R_k=r]=r+\frac{(n-r)^2-r^2}{n^2}
=1+\left(1-\frac2n\right)r.
$$
Thus $\boxed{\alpha=1,\quad\beta=1-2/n}$.
= Solution
{c}
Only upward and downward moves change the value, so the <conditional expectation> is
$$
\mathbb E[R_{k+1}\mid R_k=r]=r+\frac{(n-r)^2-r^2}{n^2}
=1+\left(1-\frac2n\right)r.
$$
Thus $\boxed{\alpha=1,\quad\beta=1-2/n}$.