Note: Please include your R codes in an appendix at the end of your answers, 1. Bob, a budding investor, wants to model the movement of stock price over time. Let X denote the stock price movement indicator at time t, given by: 1 X := { if stock price goes up, if stock price remains the same, if stock price goes down. its Bob assumes that {X} is a Markov process, i.e., the distribution of Xe conditional on history only depends on X4-1, i.e., P(X; = aX:-1, X-2, ...) = P(X= a X:-1). Further- more, Bob specifies the following joint probability distribution of Xt-1 and Xe 1 0 -1 0.1 0.05 0.1 1 X:-10 - 1 0.05 0.05 0.1 0.05 0.1 1. (a) Find the value of u (b) Find the marginal distribution of X. (c) Find the unconditional mean of X. (d) Find the unconditional variance of X. (e) Find the conditional distributions of Xt given X -1 = 1, X-1 = 0, and X:-1 = -1. (f) Find the conditional means of Xe given Xt-1 = 1, X:-1 = 0, and X:-1 = -1. (g) Find the conditional variances of Xe given X-1 = 1, X -1 = 0, and X:-1 = = -1. (h) Is (Xi} a u.n.? Prove or disprove it. (Hint: part (c)-(d) may be helpful.] (i) Is {X2} an m.d...? Prove or disprove it. (Hint: part (1) may be helpful.] G) Is {X} an ii.d. process? Prove or disprove it. (Hint: part (g) may be helpful.] Note: Please include your R codes in an appendix at the end of your answers, 1. Bob, a budding investor, wants to model the movement of stock price over time. Let X denote the stock price movement indicator at time t, given by: 1 X := { if stock price goes up, if stock price remains the same, if stock price goes down. its Bob assumes that {X} is a Markov process, i.e., the distribution of Xe conditional on history only depends on X4-1, i.e., P(X; = aX:-1, X-2, ...) = P(X= a X:-1). Further- more, Bob specifies the following joint probability distribution of Xt-1 and Xe 1 0 -1 0.1 0.05 0.1 1 X:-10 - 1 0.05 0.05 0.1 0.05 0.1 1. (a) Find the value of u (b) Find the marginal distribution of X. (c) Find the unconditional mean of X. (d) Find the unconditional variance of X. (e) Find the conditional distributions of Xt given X -1 = 1, X-1 = 0, and X:-1 = -1. (f) Find the conditional means of Xe given Xt-1 = 1, X:-1 = 0, and X:-1 = -1. (g) Find the conditional variances of Xe given X-1 = 1, X -1 = 0, and X:-1 = = -1. (h) Is (Xi} a u.n.? Prove or disprove it. (Hint: part (c)-(d) may be helpful.] (i) Is {X2} an m.d...? Prove or disprove it. (Hint: part (1) may be helpful.] G) Is {X} an ii.d. process? Prove or disprove it. (Hint: part (g) may be helpful.]