Skip to main content Contents Prev Up Next \(\renewcommand{\glspostdescription}{}
\def\cprime{\char"7E }
\def\cdprime{\char"7F }
\def\eoborotnoye{\char'013}
\def\Eoborotnoye{\char'003}
\newcommand{\beq}[1]{\begin{equation}\label{#1}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\weakstar}{weak${}^*$ }
\newcommand{\rng}{\operatorname{rng}}
\newcommand{\ex}{\operatorname{ex}}
\newcommand{\colex}{\prec_{\operatorname{colex}}}
\newcommand{\lex}{\prec_{\operatorname{lex}}}
\newcommand{\per}{\operatorname{per}}
\newcommand{\permat}{\operatorname{pm}}
\renewcommand{\hom}{\operatorname{hom}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\degree}{d}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section C.1 Exercises
Show that, for any sequence \(a_1,\dots,a_n\) of real numbers,
\begin{equation*}
\left(\sum_{i=1}^n a_i\right)^2 \leq \left(\sum_{i=1}^n|a_i|^{2/3}\right)\left(\sum_{i=1}^n|a_i|^{4/3}\right)\text{.}
\end{equation*}
Let \(X\) and \(X'\) be independent and identically distributed discrete random variables such that \(|\{x: \mathbb{P}(X=x)>0\}|=n\text{.}\) Prove that \(\mathbb{P}(X=X')\geq 1/n\text{.}\)
