For the following attachments, provide solution to all the questions and give insights where necessary.

A random sample x1, X2....,x, is taken from a population, which has the probability distribution function F(x) and the density function f(x) . The values in the sample are arranged in order and the minimum and maximum values XMy and XMAX are recorded. (i) Show that the distribution function of X MAX is [F(x)]", and find a corresponding formula for the distribution function of X MIN - The original distribution is now believed to be a Pareto(a,1) distribution, ie the probability density function is: f ( x) =- (1+ x)a+1. *20 (ii) Find the distribution function of X , and hence find the distribution function of X MAX - (iii) Show that the probability density function for the distribution of X MIN , is: fx (x) = - na x20 (1 + x) na+1 (iv) A random sample of 25 values gives a sample value for X MIN of 23. Use the distribution of X MIN to obtain a maximum likelihood estimate of a . (v) The same random sample gives a value of X MAX of 770. Obtain an equation for the maximum likelihood estimator of a using x MAY . Comment on the difficulty of solving this equation. (vi) What further information would you need here in order to obtain a method of moments estimate of o ?[15 points] Consider the following market equilibrium model. The demand and supply functions are Qd = a - bP - exp (cP) + dln(Yo) , Q= = e+fP+gp", respectively. P is the market price, Yo is income, and In () is the logarithmic function with the natural base. Assume that a, b, c, d, e, f, 9, and a are positive constants. (a) [5 points] In the equilibrium in which Q = Qa = Qs, we may set up a two-equation system: F' (P, Q; Yo) = a-bP - exp (cP) + dln (Yo) - Q =0 (20) F2 ( P, Q; Y.) = e+fP+ gp" - Q=0, (21) 3 Show that in this two-equation system the conditions of the implicit function theorem hold