series of Equations with number

The Latex code is

\begin{eqnarray}
E(y_i|z_i=1,\theta^{'}) &=& \int_t^{\infty}yf(y|y>t)dy \\
&=& \int_t^{\infty}yf(y-t)dy \\
&=& t+\theta^{'}
\end{eqnarray}

series of Equations

The Latex code is

\begin{eqnarray*}
E(y_i|z_i=1,\theta^{'}) &=& \int_t^{\infty}yf(y|y>t)dy \\
&=& \int_t^{\infty}yf(y-t)dy \\
&=& t+\theta^{'}
\end{eqnarray*}

Conditional function

The Latex code is

$$
|x| = \left\{ \begin{array}{rl}
-x &\mbox{ if $x<0$} \\
x &\mbox{ otherwise}
\end{array} \right.
$$

Double integral

The Latex code is

$\mathop{\int \!\!\! \int}_{x,y \in R^2}\langle x,y \rangle dxdy$

Matrix

The Latex code is

\[\sigma^2(Y) = \left(
\begin{array}{cccc}
\sigma^2(Y_1) & \sigma(Y_1,Y_2) & \ldots & \sigma(Y_1,Y_n) \\
\sigma(Y_2,Y_1) & \sigma^2(Y_2) & \ldots & \sigma( Y_2,Y_n) \\
\vdots & \vdots & & \vdots \\
\sigma(Y_n,Y_1) & \sigma(Y_n,Y_2) & \ldots & \sigma^2(Y_n)
\end{array}
\right)
\]

Vector

The Latex code is

\[Y = \left(
\begin{array}{c}
Y_1 \\
Y_2 \\
\vdots \\
Y_n
\end{array}
\right)
\]

combination formula

$${n+1\choose k} = {n\choose k} + {n \choose k-1}$$