2.15 Some sets of probability distributions. Let r be a real-valued random variable with prob(x = ai) = pi, i = 1, .., n, where ai < a2 < ·.. < an. Of course p E R" lies in the standard probability simplex P = {p | 1ªp= 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?) (a) a < Ef(æ) < B, where Ef(x) is the expected value of f(x), i.e., E f (x) E Pif(ai). (The function f : R → R is given.) (b) prob(x > a) < B. (c) Er*| < aE æl. (d) Ex2 < a. (e) Ex² > a. (f) var(x) a. (h) quartile(x) > a, where quartile(x) = inf{B | prob(x < B) > 0.25}. (i) quartile(x) a.
2.15 Some sets of probability distributions. Let r be a real-valued random variable with prob(x = ai) = pi, i = 1, .., n, where ai < a2 < ·.. < an. Of course p E R" lies in the standard probability simplex P = {p | 1ªp= 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?) (a) a < Ef(æ) < B, where Ef(x) is the expected value of f(x), i.e., E f (x) E Pif(ai). (The function f : R → R is given.) (b) prob(x > a) < B. (c) Er*| < aE æl. (d) Ex2 < a. (e) Ex² > a. (f) var(x) a. (h) quartile(x) > a, where quartile(x) = inf{B | prob(x < B) > 0.25}. (i) quartile(x) a.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please send step by step complete handwritten solution for Q 2.18
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) 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{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.
(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).
{(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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