  We consider a one-dimensional discrete symmetric random walk with a
reflecting boundary at the origin. Generating functions are found for the 2-
dimensional probability distribution P{Sn = x,max1?j?n Sn = a} of being at
position x after n steps, while the maximal location that the walker has
achieved during these n steps is a. We also obtain the familiar (marginal)
1-dimensional distribution for Sn = x, but more importantly that for max1?j?n
Sj = a asymptotically at fixed a2/n. We are able to compute and compare the
expectations and variances of the two one-dimensional distributions, finding
that they have qualitatively similar forms, but differ quantitatively in the
anticipated fashion.
