I'm interested in a formula to calculate the average time (step count) at which a discrete one-dimensional random walk with $n$ steps (of lengths -1 and +1) either builds its maximum or minimum excursion from the starting point. From simulation experiments I can see that the high/low seems to be approximately build after either $n \times 0.205$ (for the extremum build first) or $n \times 0.795$ steps (for the other extremum of the walk). However, how is it calculated exactly? And can this formual be derived?
Thanks a lot in advance for your help!