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