\[\definecolor{red}{RGB}{255,0,0}
\definecolor{purple}{RGB}{154,0,154}
\definecolor{blue}{RGB}{0,0,224}
\definecolor{green}{RGB}{0,187,0}
\definecolor{yellow}{RGB}{255,234,0}
\definecolor{pink}{RGB}{255,77,166}
\definecolor{orange}{RGB}{255,120,0}
\definecolor{grey}{RGB}{128,128,128}
\definecolor{black}{RGB}{0,0,0}
\definecolor{white}{RGB}{255,255,255}
\newcommand{\R}{\_mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\P}{\mathbb{P}}
\newcommand\given[1][]{\:#1\vert\:}
\DeclareMathOperator{\Corr}{Corr}
\DeclareMathOperator{\Cov}{Cov}\]