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3.24 Some functions on the probability simplex. Let x be a real-valued random variable which takes values in {a1,..., an} where a1 < a2 < .…. < an, with prob(x = a;) = pi, i = 1,..., n. For each of the following functions of p (on the probability simplex {p E R | 1"p = 1}), determine if the function is convex, concave, quasiconvex, or quasicon- cave. (а) Ех. (b) prob(x > a). (c) prob(a

3.24 Some functions on the probability simplex. Let x be a real-valued random variable which takes values in {a1,..., an} where a1 < a2 < .…. < an, with prob(x = a;) = pi, i = 1,..., n. For each of the following functions of p (on the probability simplex {p E R | 1"p = 1}), determine if the function is convex, concave, quasiconvex, or quasicon- cave. (а) Ех. (b) prob(x > a). (c) prob(a

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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please send handwritten solution for 3.24 Part a b c
3.24 Some functions on the probability simplex. Let x be a real-valued random variable which takes values in {a1,..., an} where a1 < a2 < .… < an, with prob(x = ai) = pi, i = 1,..., n. For each of the following functions of p (on the probability simplex {p E R | 1"p = 1}), determine if the function is convex, concave, quasiconvex, or quasicon- cave. (a) Ex. (b) prob(x > a). (c) prob(a <x < B). (d) E1 Pi log Pi, the negative entropy of the distribution. (e) var x = E(x – Ex)2. (f) quartile(x) = inf{3 | prob(x < B) > 0.25}. (g) The cardinality of the smallest set A C {a1,..., an} with probability > 90%. (By cardinality we mean the number of elements in A.) (h) The minimum width interval that contains 90% of the probability, i.e., inf {B – a | prob(a < x < B) > 0.9}.
Transcribed Image Text:3.24 Some functions on the probability simplex. Let x be a real-valued random variable which takes values in {a1,..., an} where a1 < a2 < .… < an, with prob(x = ai) = pi, i = 1,..., n. For each of the following functions of p (on the probability simplex {p E R | 1"p = 1}), determine if the function is convex, concave, quasiconvex, or quasicon- cave. (a) Ex. (b) prob(x > a). (c) prob(a <x < B). (d) E1 Pi log Pi, the negative entropy of the distribution. (e) var x = E(x – Ex)2. (f) quartile(x) = inf{3 | prob(x < B) > 0.25}. (g) The cardinality of the smallest set A C {a1,..., an} with probability > 90%. (By cardinality we mean the number of elements in A.) (h) The minimum width interval that contains 90% of the probability, i.e., inf {B – a | prob(a < x < B) > 0.9}.
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