Problem 5 [15 marks]

A random walk, also known as a stochastic or random process, can be described by a path that consists of a succession of random steps on some mathematical space such as the integers. A Gaussian random walk is a random walk where the random steps are normally distributed with some mean ? and some variance ?2. For this problem, we will consider Gaussian random walks, i.e. random walks with steps that come from the standard normal distribution. We can define the state Sn of the random walk at step n as follows:

n Sn = ?Xi, Xi ? N(0,1), i = 1,...,n

i=1 (a) Write a simulation in R to approximate the expected distance from the starting point, E?|Sn|?,

for n = 10, 20, . . . , 100. Use at least N =10,000 replications for each value of n. [5 marks]

2

(b) IdentifythedistributionofSn,andgivea95%confidenceintervalfor|Sn|,forn=10,20,...,100. [5 marks]

(c) Write a simulation in R to verify your theoretical confidence intervals in part (b). Use at least N =10,000 replications for each value of n. [5 marks]

Problem 5 [15 marks] A random walk, also known as a stochastic or random process, can be described by a path that consists of a succession of random steps on some mathematical space such as the integers. A Gaussian random walk is a random walk where the random steps are normally distributed with some mean ,u and some variance 02. For this problem, we will consider Gaussian random walks, i.e. random walks with steps that come from the standard normal distribution. We can dene the state 8,, of the random walk at step 11 as follows: 11. SH:ZX,-, X,~N(0,1), 2': 1,...,n i=1 (a) Write a simulation in R to approximate the expected distance from the starting point, E (|Sn|), for n = 10, 20, . . . , 100. Use at least N 210,000 replications for each value of n. [5 marks] (b) Identify the distribution of Sn, and give a 95% confidence interval for | Sn), for n = 10, 20, . .., 100. [5 marks] (c) Write a simulation in R to verify your theoretical confidence intervals in part (b). Use at least N =10,000 replications for each value of n. [5 marks]