\renewcommand{\Pr}{\mathbf{P}}                      % probability        
\newcommand{\E}{\mathbf{E}}                         % expectation
\newcommand{\I}{\mathbf{I}}                         % indicator function
\newcommand{\var}{\mathrm{var}}                     % variance
\newcommand{\cov}{\mathrm{cov}}                     % covariance
\newcommand{\cor}{\mathrm{cor}}                     % correlation

%\newcommand{\R}{\mathbb{R}}                         % reals
\newcommand{\Rp}{\mathbb{R}_+}                      % nonnegative reals
\newcommand{\Z}{\mathbb{Z}}                         % integers
\newcommand{\Zp}{\mathbb{Z}_+}                      % nonnegative integers

\newcommand{\BB}{\mathcal{B}}
\newcommand{\FF}{\mathcal{F}}
\newcommand{\GG}{\mathcal{F}}


\renewcommand{\vec}[1]{\text{\boldmath $#1$}}       % vector
\newcommand{\comp}[1]{#1^c}                         % complement
\newcommand{\gvn}{\,\arrowvert\,}                   % conditional on

\newcommand{\Exp}[1]{\mathrm{Exp}(#1)}              % exponential
\newcommand{\pois}[1]{\mathrm{Pois}(#1)}            % Poisson
\newcommand{\geo}[1]{\mathrm{Geo}(#1)}              % geometric
\newcommand{\bin}[2]{\mathrm{Bin}(#1,#2)}           % binomial
\newcommand{\unif}[2]{\mathrm{U}(#1,#2)}            % uniform
\newcommand{\norm}[2]{\mathrm{N}\left(#1,#2\right)} % normal
\newcommand{\gam}[2]{\Gamma(#1,#2)}                 % gamma
\newcommand{\bet}[2]{\beta(#1,#2)}                  % beta

\newcommand{\be}{\begin{equation}}                  % begin equation 
\newcommand{\ee}{\end{equation}}                    % end equation
