Prove that convex combination of probability measure is also a probability measure
Prove that convex combination of probability measure is also a probability measure
need help with all a, b, c
2. 15 Marks (a) Suppose that f : R" R is convex but not necessarily smooth. Prove that h-af is a (b) Suppose that f : R -R is convex and smooth. Also assume that f(x) > 0 for all z (c) Show that the set S = {(x,y) : y > 0} is convex and that the function f(x,y)-x2/v is convex function if a-0. Show with a simple example that this is...
Find a cone in R2 that is not convex, Prove that a subset X of Rr is a convex cone if and only if x,y eX implies that Xx+ py E X for all
Find a cone in R2 that is not convex, Prove that a subset X of Rr is a convex cone if and only if x,y eX implies that Xx+ py E X for all
Prove that the convex hull of a set using the fact that it is compact. x1,., nin R" is bounded,, without
Prove that the convex hull of a set using the fact that it is compact. x1,., nin R" is bounded,, without
Let P be some probability measure on sample space S = [0, 1]. (a) Prove that we must have limn→∞ P((0, 1/n) = 0. (b) Show by example that we might have limn→∞ P ([0, 1/n)) > 0.
Let g, h be two real-valued convex functions on R. Let m(x) = max{h(x), g(x)). Prove that m(x) is also convex 3.
1. (Neutral Geometry) Let DABCD be a convex quadrilateral such that AB CD and ADBC, Prove that DABCD is a parallelogram (so you must prove that AB| CD
1. (Neutral Geometry) Let DABCD be a convex quadrilateral such that AB CD and ADBC, Prove that DABCD is a parallelogram (so you must prove that AB| CD
Find the projection of vector on the convex linear
combination?
Thank You!
3 Let t = span{f}]}._ = span{{{1}+{[1]}, and let S be the set of convex linear combinations of | and [2]. For i = [!] find (a) proje V. (b) proj, v. (c) projs 7.
coherent risk measure
Prove that expected shortfall is a coherent risk measure.
Prove that expected shortfall is a coherent risk measure
Prove that each of the following sets is convex (a) {(x1, 22, x3) E R3 | 0 < 띠, x2, 23 and x1 + 2x2 + 3x3 6)
5. (a) Show that if the functions f and g are log-convex, f+g is also log-convex. Give a counter example to show that this is not true for log-concave functions (Hint: log(f +g)log(elogf +elogs). Show that this is convex by the second-order test for convexity.) (Hint: Use the definition of log-convex functions.) (Note: Harmonic mean of a,b is defined as T^T.) b) Suppose f is convex, g is non-decreasing and log-convex. Show that h(x) g(f(x)) is log-convex. (c) Show that...