Population Regression Function

To plot functions with swapped axes, you can use \addplot (function, x); instead of the normal \addplot {function};:

\documentclass[border=10pt]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.9}
\begin{document}

\begin{tikzpicture}[ % Define Normal Probability Function
declare function={
            normal(\x,\m,\s) = 1/(2*\s*sqrt(pi))*exp(-(\x-\m)^2/(2*\s^2));
        }
       ]
\begin{axis}[
    no markers,
    domain=-3.2:3.2,
    samples=100,
    axis lines=left,
    enlarge x limits=true,
    xtick={0,0.5,1},
    xmajorgrids,
    xticklabels={},
    ytick=\empty,
    xticklabels={$x_1$, $x_2$, $x_3$},
    xlabel=$x$, xlabel style={at=(xticklabel cs:1), anchor=south},
    ylabel=$y$, ylabel style={at=(yticklabel cs:1), rotate=-90, anchor=east},
  ]

\addplot [samples=2, domain=0:1.4] {1.5*x};
\addplot[cyan!50!black, thick] ({normal(x, 0, 1)},x);
\addplot[cyan!50!black, thick, domain=-2.5:4.49] ({normal(x, 1.2, 1.5)+0.5},x);
\addplot[cyan!50!black, thick, domain=-1:4.4] ({normal(x, 2, 0.75)+1},x);

\end{axis}

\end{tikzpicture}
\end{document}