Consider a random walk \(S_i\), which begins at zero, and makes \(2n\) steps.
Let \(k < n\). Show that

1. \(\mathbf{E}\Big( |S_{2k}| \  \Big|\  |S_{2k−1}| = r\Big) = r\);
2. \( \mathbf{E}\Big(|S_{2k+1}| \ \Big|\  |S_{2k}| = r\Big) = 1 \text{ if  } r = 0, \text{ and } \mathbf{E}\Big(|S_{2k+1}| \ \Big|\  |S_{2k}| = r\Big) = r\text{  otherwise}.\)