random walk

Questions 1. A coin is tossed indenitely. Let p E ]0, 1]\\{1/2} be the probability of obtaining heads at one toss. Let Sn be the random variable equal to the number of heads less the number of tails in n tosses. We set 50 = 0. i. Show that Sn is a simple random walk. ii. Compute P(Sn = 0), depending on the parity of 11. iii. Let N = |{n E N *1 Sn 2 0} be the number of times when the trajectory cuts the time axis. Show that the expected value of N is nite. iv. Compute P(lim(Sn = 0)). E