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
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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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