Some sets of probability distributions. Let x be a real-valued random variable with prob(x=ai)=pi, i=1,…,n, where a1<a2<⋯<an. Of course p∈Rn lies in the standard probability simplex P={p∣1Tp=1,p⪰0}. Which of the following conditions are convex in p? (That is, for which of the following conditions is the set of p∈P that satisfy the condition convex?)
(a) α≤Ef(x)≤β, where Ef(x) is the expected value of f(x), i.e., Ef(x)=∑i=1npif(ai). (The function f:R→R is given.)
(b) prob(x>α)≤β.
(c) E∣x3∣≤αE∣x∣.
(d) Ex2≤α.
(e) Ex2≥α.
(f) var(x)≤α, where var(x)=E(x−Ex)2 is the variance of x.
(g) var(x)≥α.
(h) quartile(x)≥α, where quartile(x)=inf{β∣prob(x≤β)≥0.25}.
(i) quartile(x)≤α.