Table 1.

Left column: mathematical expressions of the probability density functions (pdf's) for RTs for a single boundary diffusion model, two boundary diffusion model, and the reciprocal Normal.

Time-domain	Rate-domain
Inverse Gaussian	Reciprocal inverse Gaussian
$IG(\mu ,{\sigma }^2)={\left(\frac{{\mu }^3}{2\pi {\sigma }^2{\chi }^3}\right)}^{\frac{1}{2}}\exp \left[\frac{-\mu {(\chi -\mu )}^2}{2{\sigma }^2\chi }\right]$	$recIG(\mu ,{\sigma }^2)={\left(\frac{{\mu }^3}{2\pi {\sigma }^2\chi }\right)}^{\frac{1}{2}}\exp \left[\frac{2{\mu }^2-\mu /\chi -\chi {\mu }^3}{2{\sigma }^2}\right]$
First passage time distribution for the boundary of the two boundaries a and b for the pure DDM with diffusion constant, s	Reciprocal first passage time distribution boundary a of the two boundaries a DDM
$\begin{array}{l}B(\xi ,a,b,z)=\frac{\pi s^2}{{(a-b)}^2}\exp \left[\frac{\xi (a-z)}{s^2}-\frac{{\xi }^{2}t}{2s^2}\right]\hfill \\ {\displaystyle \sum _{k=1}^{\infty }k}\exp \left[-\frac{k^2{\pi }^2{s}^{2}t}{2{(a-b)}^2}\right]\sin \left[\frac{k\pi (a-z)}{a-b}\right]\hfill \end{array}$	$\begin{array}{l}recB(\xi ,a,b,z)=\frac{\pi s^2}{t^2{(a-b)}^2}\exp \left[\frac{\xi (a-z)}{s^2}-\frac{{\xi }^2}{2t{s}^2}\right]\hfill \\ {\displaystyle \sum _{k=1}^{\infty }k}\exp \left[-\frac{k^2{\pi }^2{s}^2}{2t{(a-b)}^2}\right]\sin \left[\frac{k\pi (a-z)}{a-b}\right]\hfill \end{array}$
see Ratcliff and Smith (2004)
Reciprocal truncated Normal	Truncated Normal
$rectrN(\mu ,\sigma )=\frac{\exp \left[-\frac{{(1/t-\mu )}^2}{2{\sigma }^2}\right]}{\sqrt{2\pi }\left(1-\varphi \left(-\frac{\mu }{\sigma }\right)\right)\sigma t^2}$	$trN(\mu ,\sigma )=\frac{\exp \left[-\frac{{(t-\mu )}^2}{2{\sigma }^2}\right]}{\sqrt{2\pi }\left(1-\phi \left(-\frac{\mu }{\sigma }\right)\right)\sigma }$

Φ = Normal cdf; ξ = drift rate; a = upper boundary; b = lower boundary; z = starting point. Right column: equivalent pdf's in the rate (reciprocal RT) domain. See Harris and Waddington (2012) for the mathematical relationship between the two domains.