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To get an the average distance from zero after N steps, the following argument was used: The distance after one step is 1. If after some steps, the distance is D, then with the next step the distance is either D-1 or D+1. The squares of the distances after the next step are either D^2-2D+1 or D^2+2D+1. Since both are equally likely, the change in distance is just the average of them. Adding them up and dividing by two, to take the arithmetic mean. the new distance squared is D^2+1. Since the square of the distance is 1 after the first step and increases by one every step, therefore the distance after N steps is the square root of N, on average when both jumps happen as often.

Considering the case after one step, since both jumps are as likely, to zero and two, the same argument says that the average distance is the square root of two. But why is that, since jumping to 0 or 2 is just as likely, they happen as often and therefore the average distance should be one.

Instead shouldn't it be said that since the distance is either 0 or 2, we should take the average of them and not of the squares to get the average distance?

What is the reasoning for using the squares? If I square 9 and 11 and take the square root of their average, I don't get 10.