(use R) 4. [15 marks in total] Consider a group of 200 people, 100 of whom at the moment support the Labour Party and 100 of whom support the Conservative Party. Assume that the probability that a Labour supporter changes to a Conservative supporter any month is 0.015, while the probability that a Conservative supporter changes to a Labour supporter is 0.01.

a. [4 marks] Introduce the random variables for i = 1, . . . , 100:

X (1) i = ( 1, Labour supporter number i still supports Labour at the end of month 1

0 Labour supporter number i changes to a Conservative supporter at the end of month 1,

and

X (2) i = ( 1, Conservative supporter number i changes to a Labour supporter at the end of month 1

0 Conservative supporter number i still supports Conservative at the end of month 1.

Using these random variables express the number of Labour Party supporters at the end of the first month, then find the expected number of Labour supporters at the end of the first month. Justify every step of your argument.

b. [4 marks] Define random variables X (1) i , X (2) i , i = 1, . . . , 100 whose sum gives you the number of Conservative Party supporters at the end of the first month. What is the expected number of Conservative Party supporters at the end of the first month?

c. [4 marks] Suppose we want to simulate how the number of Conservative and Labour Party supporters changes over time. Write an R code that computes and stores the number of Labour and Conservative Party supporters each month for the next 10 years.

d. [3 marks] Plot how the number of Conservative and Labour Party supporters changes over time.