MathDB
Operations on maps on random variables

Source: Alibaba Global Math Competition 2021, Problem 4

July 4, 2021
random variablescollege contests

Problem Statement

Let (Ω,A,P)(\Omega, \mathcal{A},\mathbb{P}) be a standard probability space, and X\mathcal{X} be the set of all bounded random variables. For t>0t>0, defined the mapping RtR_t by R_t(X)=t\log \mathbb{E}[\exp(X/t)],   X \in \mathcal{X}, and for t∈(0,1)t \in (0,1) define the mapping QtQ_t by Q_t(X)=\inf\{x \in \mathbb{R}: \mathbb{P}(X>x) \le t\},   X \in \mathcal{X}. For two mappings f,g:X→[−∞,∞)f,g:\mathcal{X} \to [-\infty,\infty), defined the operator □\square by f\square g(X)=\inf\{f(Y)+g(X-Y): Y \in \mathcal{X}\},   X \in \mathcal{X}. (a) Show that, for t,s>0t,s>0, Rt□Rs=Rt+s.R_t \square R_s=R_{t+s}. (b) Show that, for t,s>0t,s>0 with t+s<1t+s<1, Qt□Qs=Qt+s.Q_t \square Q_s=Q_{t+s}.
Back to ProblemsView on AoPS