Question

LSM 5 Part C. Suppose that you are walking on a straight line. You start at position X, -0, and only walk in the positive direction. Your positions afcer taking the ith step is denoted by X,. For each step, your step size, denoted by S, = X,-X,-ı , is a random variable uniformly distributed between 1 foot and 2 eet. Assume that sizes of different steps are mutually independent. 8. (10 credits) Let Xy be your position after taking N steps where Nis a given constant. Obtain EXy] and Var[X) 9. (10 credits) Use Chebyshev's inequality to obtain an upper bound to Now assume that in each step, you also make a random decision on the direction of your step. With a probability of 0.5, you will take a step in the positive direction; and with a probability of 0.5, you will take a step in the negative direction. Each step size is still random and uniformly distributed between 1 foot and 2 feet. Suppose that your direction decisions are mutually independent and also independent from everything else.

Answer 1

lo iu a ienstant Sj 2. 25 e(St) 2 5 1 2 umg C