Consider an insect that starts flying from the origin at time 0. At time1, it flies to the random point (X1,Y1). Starting from this point as the origin, it flies to a random point (X2,Y2), and so on. (i) Consider the co-ordinates of the fly at time n relative to its position at time 0. Assuming thatn= 100 and that the (Xi, Yi) are i. i.d. with each co-ordinate having a standard normal distribution, find the probability that the insect is at 50 units above the y=x line (relative to the origin at time 0).
(ii) Consider the total distance that the fly has actually flown starting from time 0. Under the assumptions of (i), find an approximation to the probability that this quantity is between 75 and 150.