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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}.

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}.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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2.15 Some sets of probability distributions. prob(x = ai) = 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 E 0}. Which of the following conditions are convex in p? (That is, for which of the following conditions is the set of p E P that satisfy the condition convex?) Let x be a real-valued random variable with (a) a < Ef(æ) < B, where E f(x) is the expected value of f(x), i.e., E f(x) E Pif(a;). (The function f : R → R is given.) (b) prob(r > a) S B. (c) E 2°| < a E |r|. (d) Ex² < 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{B| 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" × R++. 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 vi E R" and t; > 0. (a) The polyhedron C = (b) The hyperplane C = {(v, t) | f"v+ gt = h} (with f and g not both zero). (c) The halfspace C = {(v, t) | f"v+ gt < h} (with f and g not both zero). {(v, t) | Fv + gt < h}. (d) The polyhedron C = 2.18 Invertible linear-fractional functions. Let f : R" → R" be the linear-fractional function f(x) = (Ax + b)/(c"x+ d), dom f = {r | c" x +d>0}. Suppose the matrix A Q = b. d is nonsingular. Show that f is invertible and that f is a linear-fractional mapping. Give an explicit expression for f-1 and its domain in terms of A, b, c, and d. Hint. It may be easier to express f in terms of Q.
Transcribed Image Text:2.15 Some sets of probability distributions. prob(x = ai) = 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 E 0}. Which of the following conditions are convex in p? (That is, for which of the following conditions is the set of p E P that satisfy the condition convex?) Let x be a real-valued random variable with (a) a < Ef(æ) < B, where E f(x) is the expected value of f(x), i.e., E f(x) E Pif(a;). (The function f : R → R is given.) (b) prob(r > a) S B. (c) E 2°| < a E |r|. (d) Ex² < 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{B| 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" × R++. 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 vi E R" and t; > 0. (a) The polyhedron C = (b) The hyperplane C = {(v, t) | f"v+ gt = h} (with f and g not both zero). (c) The halfspace C = {(v, t) | f"v+ gt < h} (with f and g not both zero). {(v, t) | Fv + gt < h}. (d) The polyhedron C = 2.18 Invertible linear-fractional functions. Let f : R" → R" be the linear-fractional function f(x) = (Ax + b)/(c"x+ d), dom f = {r | c" x +d>0}. Suppose the matrix A Q = b. d is nonsingular. Show that f is invertible and that f is a linear-fractional mapping. Give an explicit expression for f-1 and its domain in terms of A, b, c, and d. Hint. It may be easier to express f in terms of Q.
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