Provide solutions to the following questions.

The random variables X1..... X,, represent independent observations from a population with mean / and variance of. (i) State the mean and variance of the sample mean X = - X;. [2] (ii) Show that _(X; - X)? = Ex? -nx2. [2] (iii) Determine the expected value of E (X; - X). [3] (iv) Hence show that E($2) =02, where $2 = - 1 n-1 2 x? - nx2. [1] (v) For the special case where the parent population is normally distributed and the values of the population parameters / and o are known, state two results that specify the precise distributions of X and E( X, - X) . [2] [Total 10]Given the following equation, derive the expression for the IS curve. C = 100 + 0.8Y (Consumption function) I = 10-10r (Investment function) G = 10 (Government purchases) T - 0.25 (Tax rate)