\sum_{<text>}

# Add this code before \begin{document} 

\usepackage{eso-pic}
\newcommand\BackgroundPic{
\put(0,0){
\parbox[b][\paperheight]{\paperwidth}{%
\vfill
\centering
\includegraphics[width=\paperwidth,height=\paperheight,
keepaspectratio]{background.png}%
\vfill
}}}

# and this immediately after \begin{document} :

\AddToShipoutPicture{\BackgroundPic}

Y_i \sim \left\{ \begin{array}{l}
0, \text{with probability } (1-\psi) \\
\text{Poisson}(\mu), \text{with probability } \psi
\end{array}

% bold face in formulas
\def\mbf#1{\mathchoice{\hbox{\boldmath $\displaystyle #1$}}
        {\hbox{\boldmath $\textstyle #1$}}
        {\hbox{\boldmath $\scriptstyle #1$}}
        {\hbox{\boldmath $\scriptscriptstyle #1$}}}