2.15 Some sets of probability distributions. prob(x = a¡) = pi, i = 1,..., n, where ai < a2 < ·… < an. Of course p E R" lies in the standard probability simplex P = {p| 1"p = 1, p> 0}. Which of the following conditions are convex in p? (That is, for which of the following conditions is the set of pE P that satisfy the condition convex?) Let x be a real-valued random variable with (a) a < Ef(æ) < B, where Ef(x) is the expected value of f(x), i.e., E f(x) E Pif (a;). (The function f : R → R is given.) %3D (b) prob(r > a) S B. (c) E 2°| < a E |r|. (d) Ex? < a. (e) Ex² > a. (f) var(x) < a, where var(x) = (g) var(x) > a. E(x – Ex)2 is the variance of x. (h) quartile(x) > a, where quartile(x) = inf{3| prob(x < B) > 0.25}. (i) quartile(x) a. Operations that preserve convexity 2.16 Show that if S1 and S2 are convex sets in Rm+n, then so is their partial sum = {(x, y1 + y2) | x € R", y1, y2 € R", (x, Y1) E S1, (x, y2) E S2}. 2.17 Image of polyhedral sets under perspective function. In this problem we study the image of hyperplanes, halfspaces, and polyhedra under the perspective function P(x, t) = x/t, with dom P = R" x R4+. For each of the following sets C, give a simple description of P(C) = {v/t| (v,t) € C, t> 0}. conv{(v1,t1),..., (VK,tK)} where vị E R" and ti > 0. (b) The hyperplane C = {(v, t) | f"v+ gt = h} (with f and g not both zero). (a) The polyhedron C = (c) The halfspace C = {(v, t) | f"v+ gt < h} (with f and g not both zero). (d) The polyhedron C = {(v, t) | Fv + gt < h}.