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## 17.2 Wiener First Passage Time Distribution

### 17.2.1 Probability Density Function

If $$\alpha \in \mathbb{R}^+$$, $$\tau \in \mathbb{R}^+$$, $$\beta \in [0, 1]$$ and $$\delta \in \mathbb{R}$$, then for $$y > \tau$$, $\text{Wiener}(y|\alpha, \tau, \beta, \delta) = \frac{\alpha^3}{(y-\tau)^{3/2}} \exp \! \left(- \delta \alpha \beta - \frac{\delta^2(y-\tau)}{2}\right) \sum_{k = - \infty}^{\infty} (2k + \beta) \phi \! \left(\frac{2k \alpha + \beta}{\sqrt{y - \tau}}\right)$ where $$\phi(x)$$ denotes the standard normal density function; see (Feller 1968), (Navarro and Fuss 2009).

### 17.2.2 Sampling Statement

y ~ wiener(alpha, tau, beta, delta)

Increment target log probability density with wiener_lpdf( y | alpha, tau, beta, delta) dropping constant additive terms.

### 17.2.3 Stan Functions

real wiener_lpdf(reals y | reals alpha, reals tau, reals beta, reals delta)
The log of the Wiener first passage time density of y given boundary separation alpha, non-decision time tau, a-priori bias beta and drift rate delta

### 17.2.4 Boundaries

Stan returns the first passage time of the accumulation process over the upper boundary only. To get the result for the lower boundary, use $\text{wiener}(y | \alpha, \tau, 1 - \beta, - \delta)$ For more details, see the appendix of Vandekerckhove and Wabersich (2014).