Questions

1.

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)).