Solution
Random Walk Returns
Show the problem again
A particle starts at 0 on the integer number line. At each step it moves +1 or -1 with equal probability. What is the expected number of steps to return to 0 for the first time?
Worked solution
The expected return time is infinite. Although a 1D symmetric random walk is recurrent (it returns to 0 with probability 1), the expected time to do so is infinite. This is a classic result distinguishing recurrence from positive recurrence.
Source: Polya's recurrence theorem for random walks (1921). Statement written for AxiomIQ.