Since you cant set the size of math notation I was wondering if we can scale the image to be larger so that math appears smaller?
Also I`d like to be able to change the x-axis and y-axis of the subplot to be similar to to the sample image?
Finally, this might be difficult but its possible to add some sample curves/paths in light gray for the distributions that roughly match the mean and variance with some random curvature similar to the example below?
To increase the size of the plot, you just may adjust width and height. The second part of the question I don't understand. Indicating mean and variance is straightforward.
\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\pgfmathsetmacro{\offset}{0.05}
\begin{tikzpicture}[font=\sffamily,
declare function={gauss(\x,\y,\z)=\offset+1/(\y*sqrt(2*pi))*exp(-((\x-\z)^2)/(2*\y^2));}]
\begin{axis}[samples=101,smooth,hide axis,width=20cm,height=11cm]
\addplot [domain=-3:3] ({gauss(x,0.8,0)},x);
\addplot [domain=-3:3] ({1+gauss(x,1.2,0)},1+x);
\addplot [domain=-3:3] ({2+gauss(x,0.6,0)},x);
\foreach [count=\Y starting from 0] \vari/\DeltaX in {0.8/0,1.2/1,0.6/0}
{\typeout{\vari/\DeltaX}
\edef\temp{\noexpand\draw[gray] ({\Y+\offset},{\vari+\DeltaX}) -- ({\Y+gauss(\vari,\vari,0)},{\vari+\DeltaX})
({\Y+\offset},{-\vari+\DeltaX}) -- ({\Y+gauss(\vari,\vari,0)},{-\vari+\DeltaX})
(\Y+\offset,\DeltaX) -- ({\Y+gauss(0,\vari,0)},\DeltaX)
;}
\temp}
\draw[-latex] (0,-3) -- (0,3) coordinate[pos=0.4](x1) coordinate[pos=0.5] (y1)
node[below right,rotate=-90]{$x(t_i)$};
\draw[-latex] (0,-3) -- (0.5,-3) node[below left,rotate=-90]{$f\bigl(x(t_i)\bigr)$};
\draw (1,-2) -- (1,4) coordinate[pos=0.6](x2) coordinate[pos=0.5] (y2);
\draw (2,-3) -- (2,3) coordinate[pos=0.6](x3) coordinate[pos=0.5] (y3);
\addplot[-latex] coordinates{(-0.5,-4) (3,-4)};
\path (0,-4) coordinate (z1) (1,-4) coordinate (z2) (2,-4) coordinate (z3);
\coordinate (t) at (3,-4.1);
\coordinate (xi) at (-0.6,4);
\addplot[-latex] coordinates{(-0.5,-4) (3,-4)};
\addplot[-latex] coordinates{(-0.5,-4) (-0.5,4)};
\end{axis}
\foreach \X in {1,2,3}
{\fill (x\X) circle (2pt);
\draw ([xshift=-1mm]y\X) -- ([xshift=1mm]y\X);
\draw ([yshift=1mm]z\X) -- ([yshift=-1mm]z\X) node[below] {$\X$};}
\node[anchor=north east] at (t) {time};
\node[anchor=north east] at (xi) {$x(t)$};
\draw[blue,thick,shorten >=2mm,shorten <=2mm] (x1) -- (x2);
\draw[blue,thick,shorten >=2mm,shorten <=2mm] (x2) -- (x3);
\end{tikzpicture}
\end{document}
Alternative you may just decrease the size of the font.
\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\pgfmathsetmacro{\offset}{0.05}
\begin{tikzpicture}[font=\sffamily\small,
declare function={gauss(\x,\y,\z)=\offset+1/(\y*sqrt(2*pi))*exp(-((\x-\z)^2)/(2*\y^2));}]
\begin{axis}[samples=101,smooth,hide axis,width=15cm,height=8cm]
\addplot [domain=-3:3] ({gauss(x,0.8,0)},x);
\addplot [domain=-3:3] ({1+gauss(x,1.2,0)},1+x);
\addplot [domain=-3:3] ({2+gauss(x,0.6,0)},x);
\foreach [count=\Y starting from 0] \vari/\DeltaX in {0.8/0,1.2/1,0.6/0}
{\edef\temp{\noexpand\draw[gray] ({\Y+\offset},{\vari+\DeltaX}) -- ({\Y+gauss(\vari,\vari,0)},{\vari+\DeltaX})
({\Y+\offset},{-\vari+\DeltaX}) -- ({\Y+gauss(\vari,\vari,0)},{-\vari+\DeltaX})
(\Y+\offset,\DeltaX) -- ({\Y+gauss(0,\vari,0)},\DeltaX)
;}
\temp}
\begin{scope}[thick]
\draw[-latex] (0,-3) -- (0,3) coordinate[pos=0.4](x1) coordinate[pos=0.5] (y1)
node[below right,rotate=-90]{$x(t_i)$};
\draw[-latex] (0,-3) -- (0.5,-3) node[below left,rotate=-90]{$f\bigl(x(t_i)\bigr)$};
\draw[-latex] (1,-2) -- (1,4) coordinate[pos=0.6](x2) coordinate[pos=0.5] (y2);
\draw[dashed] (0,-3) -- (0,-3.5);
\draw[-latex] (2,-3) -- (2,3) coordinate[pos=0.6](x3) coordinate[pos=0.5] (y3);
\draw[dashed] (1,-2) -- (1,-3.5);
\path (0,-4) coordinate (z1) (1,-4) coordinate (z2) (2,-4) coordinate (z3);
\draw[dashed] (2,-3) -- (2,-3.5);
\coordinate (t) at (3,-4.1);
\coordinate (xi) at (-0.6,4);
\addplot[-latex] coordinates{(-0.5,-3.5) (3,-3.5)};
\addplot[-latex] coordinates{(-0.5,-3.5) (-0.5,3.5)};
\end{scope}
\pgfplotsinvokeforeach{1,...,20}{
\draw[very thin] plot[smooth] coordinates
{(0,{sign(2*rand-1)*(exp(-rand/0.8)-1)})
(1,{1+sign(2*rand-1)*(exp(-rand/1.6)-1)})
(2,{sign(2*rand-1)*(exp(-rand/1.2)-1)})};}
\end{axis}
\foreach \X in {1,2,3}
{\fill (x\X) circle (2pt);
\draw ([xshift=-1mm]y\X) -- ([xshift=1mm]y\X);
\draw ([yshift=1mm]z\X) -- ([yshift=-1mm]z\X) node[below] {$\X$};}
\node[anchor=north east] at (t) {time};
\node[anchor=north east] at (xi) {$x(t)$};
\draw[blue,thick,shorten >=2mm,shorten <=2mm] (x1) -- (x2);
\draw[blue,thick,shorten >=2mm,shorten <=2mm] (x2) -- (x3);
\end{tikzpicture}
\end{document}
