6.2 Exercise 17

A marker is placed at a reference position on a line (taken to be the origin); a coin is tossed repeatedly. If a head turns up, the marker is moved one unit to the right; if a tail turns up, the marker is moved one unit to the left.

1. Show that the position at the end of ten tosses is given by the random variable
   $X = \sum_{i = 1}^{10} I_{E_{i}} - \sum_{i = 1}^{10} I_{E_{i}^{c}} = 2 \sum_{i = 1}^{10} I_{E_{i}} - 10 = 2 S_{10} - 10$
   where E_i is the event of a head on the ith toss and S₁₀ is the number of heads in ten trials.
2. After ten tosses, what are the possible positions and the probabilities of being in each?