A coin is tossed indefinitely. Let ]0,1]\{1/2} be the probability of obtaining heads at one toss. Let be the random variable equal to the number of heads less the number of tails in n tosses. We set 0=0.

i. Show that is a simple random walk.

ii. Compute ( = 0), depending on the parity of .

iii. Let = |{ = 0} be the number of times when the trajectory cuts the time axis. Show that the expected value of is finite.

iv. Compute (( = 0)).