Two regressions: $$ y_{1i}=x_i\beta+u_{1i} $$ $$ y_{2i}=z_i\gamma+u_{2i} $$ Two errors are correlated. $x_i$ and $z_i$ I suppose are not necessarily independent (but it is not so clear from my lecture notes).
Some background: The two regressions are used to correct sample selection bias, and used in the Heckman model.
The notes say to get the following conditional probability: \begin{align} f(y_{1i},z_i\gamma+u_{2i}>0|x_i, z_i ) &= f(y_{1i}|x_i, z_i)Pr(z_i\gamma+u_{2i}>0|y_{1i}, x_i, z_i) \\ &= f(y_{1i}|x_i)Pr(z_i\gamma+u_{2i}>0|u_{1i}, x_i, z_i) \\ &=f(u_{1i})\int^{\infty}_{-z_i\gamma}f(u_{2i}|u_{1i})du_{2i} \end{align} I don't understand how $z_i$ disappear $f(y_{1i}|x_i)$, and why $f(y_{1i}|x_i)=f(x_i\beta+u_{1i}|x_i)=f(u_{1i})$.