LSM 5 Part C. Suppose that you are walking on a straight line. You start at position Xo , and only walk in the positive direction. Your positions after taking the ith step is denoted by X,. For e...
LSM 5 Part C. Suppose that you are walking on a straight line. You start at position Xo , and only walk in the positive direction. Your positions after 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 feet. Assume that sizes of different steps are mutually independent 8. (10 credits) Let X, be your position after taking N steps where N is a given constant. Obtain E] and Varlx ]. 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. 10. (10 credits) Let Xy be your position after taking N steps where N is a given constant. Obtain Ex] and Var[x 11. (10 credits) For large N, use the Central Limit Theorem to approximate Plxy - EX,2 /N. Please express your result using the Q) function.
LSM 5 Part C. Suppose that you are walking on a straight line. You start at position Xo , and only walk in the positive direction. Your positions after 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 feet. Assume that sizes of different steps are mutually independent 8. (10 credits) Let X, be your position after taking N steps where N is a given constant. Obtain E] and Varlx ]. 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. 10. (10 credits) Let Xy be your position after taking N steps where N is a given constant. Obtain Ex] and Var[x 11. (10 credits) For large N, use the Central Limit Theorem to approximate Plxy - EX,2 /N. Please express your result using the Q) function.