Question 6 [Total: 22 marks] Political conflicts play a significant role in affecting the stock market performance. Let W be a dummy variable such that W = 1 if there is a war in the coming month and W = 0 if there is no war. Let X be the market return in the coming month. Carol assumes the following model. The probability of a war in the coming month is 0.3. If a war occurs, the market return follows a normal distribution with mean -0.2 and variance 0.04; if a war does not occur, the market return follows a normal distribution with mean 0.1 and variance 0.01. The assumptions are summarised in the table below. Probability Distribution of X |W W=1 0.3 N(-0.2, 0.04) W=0 0.7 N(0.1, 0.01) (a) [2 marks] Are W and X independent? Explain briefly. (b) [3 marks] Suppose a war occurs in the coming month. What is the probability that the stock market will suffer a negative return? (c) [4 marks] Suppose the stock market suffers a negative return. What is the probability that a war occurs? (d) [3 marks] Find E(X). (e) [4 marks] Compute E(X' [ W = 1) and E(X? | W = 0). (f) [3 marks] Using parts (d) and (e), compute E(X?) and Var(X). (g) [3 marks] Let W, denote the war status in month t, and let X, denote the stock market return in month t. Suppose the joint distribution of W, and X are independent and identically distributed according to the above table across all the 12 months of 2022. Is the time series { X } strictly stationary? Why or why not