Problem of the Week 1236

A Shrinking Random Walk

Consider a random walk in the plane that starts at the origin and moves only in the positive x and positive y directions. The direction choice at each step is governed by the flip of a fair coin. The length of the first move is √2; the length of the second move is √2/2; the length of the third move is √2/4; and so on.

At the end of infinitely many steps, what is the expected distance from the origin?

Source: Tom Yuster, Math Horizons, April 2017, pp. 32-33.

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