Answer Happy

Question 4: (People immigration) Suppose that people immigrate into a territory at a Poisson process with rate 1 = 1 per day. 1. Write the definition of a standard Poisson process. 2. What is the expected time until the tenth immigrant arrives? 3. Find the probability that the elapsed time between the tenth and the eleventh arrival exceeds 2 days. Question 5: (Bicycle race) In a bicycle race between two competitors, let Y(t) denote the amount of time (in seconds) by which the racer that started in the inside position is ahead when 100 percent of the race has been completed, and suppose that {Y(t), 2 0} can be effectively modeled as a standard Brownian motion process. 1. Write the definition of a standard Brownian motion. 2. What is the distribution of y(t) - Y(s)? 3. If the inside racer is leading by 1 seconds at the midpoint of the race, calculate P(Y(1) >01Y) = 1) the probability that she is the winner?