Question 4 [14 points] Consider a first price auction with two buyers 1,2. Their valuations v1, V2 are independent and each follows the triangle distribution in the interval [0, 2], as shown below. Diagram 1 f(x) Vi 0 X 2 (a) [3 points] For any x between 0 and 2, determine the expression of the density function f (x) of the triangle distribution given above. (b) [3 points] For any x between 0 and 2, determine the expression of the distribution function F(x) of the triangle distribution given above. (c) [4 points] The bid functions of buyers 1,2 are by = kivi, by = kyv2 where ki, k2 are positive numbers. Showing all steps of your work, show that the expected net income of buyer 1 is vi ki (1 - ki) (d) [4 points] Using the arithmetic mean-geometric mean (AM-GM) inequality, determine the choice of k, that maximizes the expected net income of buyer 1